On how many routes an aircraft is used on a typical day?

Actually, a commercial airliner pode do almost qualquer set of legs, assuming those are technically possible (range/payload/laws like ETOPS...) But the main fact that make it possible is demanda. If you run an airline, you won't make routes that don't have demand to make profit.

Aviation is a complex world. To make profit, you have to consider a massive load of parameters. You can't for example position complete set of crew in every city/airport hotel of your airline routes then decide to create fantasy flights "a la carte". Crews can't fly continuously aswell, so you can't put crew in a 737 and make them non-stop flights around the country an entire week.

One main thing you have to keep in mind : It's not one or several persons who decide "I want to build an airline with X aircraft on Y routes". It's one A-B route that decide how many passengers are available on, then, an airline has the opportunity to operate that route (and struggle to negociate all the authorizations)

Here are some hints about how airliners deals with their environment around the world :

1. What kind of flights does the airliner, and what airframes does it have in its fleet ?

   • An airliner can start with one single aircraft, says a DHC6 Twin Otter, and makes regular flights between two airports in the middle of the ocean. You can have 10, 15, maybe 20 round trip a day, some well after sunset.
   • An airliner can have half a dozen ATR or Dash 8, and a set of routes with the proper demand. You can have routes like hub-A-hub, hub-B-hub, when the demand is enough to make each flight profitable. Sometimes, you can't have enough passenger a day on each route, so the airliner makes hub-A-B-C-D-hub the day one, then hub-D-C-B-A-hub on day two, and cycle the sets...
   • An airliner with dozens of (the same) aircraft (737 or A32x) can either have the same round trip hub-X-hub destinations every day, several times a day, airlines like Air France, Southwest, Lufthansa, Air China... or the same fleet, but much more destinations operated two or three times a week : hub-A-B-hub, hub-C-D-E-hub, hub-F-G-H-nightstop-G-F-hub... like Angola Airlines (domestic and regional)
   • Long range flights (7 hours or more) requires the appropriate aircraft. If your airline can buy/lease such an airliner, the route hub-destination-hub can be operated in the limits of the demand. When the demand is enough to fill the plane along the week, then go ahead. Otherwise, the airline has to decide wether a) leave the plane on the tarmac, b) find a nearby complementary route : hub-A-B-hub, hub-B-A-hub, c) use a smaller and cheaper aircraft (if the crew is qualified on) d) combine destination A with a complementary destination B that fills schedules along the week (or bi-week) hub-A-hub-B-hub, e) replace the route with one better, f) negociate with one or several airliners to sell seats to them to fill the aircraft, g) negociate with one or several airliners operating on the destination airport to take advantage of the hubbing : make your airline land at the destination during rush hours.
   • Very long range flights can't be operated every day with one single aircraft. And very long range flights are a niche : the hours spent in that closed tube discourage passengers from taking such flights. And very long range flights are a hell to schedule : the aircraft does 3 or 4 flights a week, and stays on the tarmac the rest of the time.

2. The size of the airline :

   • If your airline is a big one, like British Airways or Japan Airlines or United, your reputation and historical operation is enough to get demand on most usual routes (and your staff has the expertise to select profitable ones) Most of your flights will be the type "hub-X-hub", either domestic, regional, or international.
   • If your airline is above average (like Kenya Airways, Aeromexico, Air India) you'll have a mix of "hub-X-hub" flights and some complementary legs to fix schedules and aircraft operation ratio : hub-A-B(-C)-Hub
   • If your airline is below average (Blue Panorama, Air Kazakstan, Belavia) chances are great to have charter flights, multiple flight shifts.. Timetables greatly vary from one year to another.
   • Small airlines like Air Austral, Republic Express (Indonesia), Insel Air will have historical regular flights, but most of them will be of type "hub-A-B-hub" and/or "hub-A-C-E-C-B-hub". Timetables also vary from one year to another, but at a less rate than above because the airline don't have enough aircraft to think of new or better routes to operate, as the ones available cannot be closed or replaced.

3. The region of operation - the airport statistics (passenger volume per year) :

   • There are regions of high frequency of flights : Western Europe, United States, Japan, South-East Asia, Eastern Australia, the Dubai-Europe axis... In those regions, you have several hundreds of round trip routes that can be operated ten times a day using a Boeing 747. Cheap flights, historical demand, dynamic business. You can operate flights of type "hub-destination-hub".
   • There are regions with much less traffic, but enought for one airline to operate the simpliest flights hub-dest-hub. However, once another airline or more decide to compete on the same route, everyone is forced to reconsider the way it operates their fleet.
   • You also have regions in the world, especially in Africa and some non-democratic countries out there, where only one airline is allowed to operate a given route. Usually, those airlines don't care about competition, as the state/country itself protect them. Ticket prices on those routes are unusually high, because they can. So the demand on those routes NEVER take off, thus, making the flights on them the same along the years. You'll have destinations operated once a week (if not once a month) You'll have very small airlines and you don't understand why, having the monopoly on the network, they are not able to grow, grow the traffic, and propose better schedules for custommers than the absurd routes and departure/arrival times. Simple : they simply don't care.

4. Night Flights - jet lag.

   • You can't just make an aircraft take off, and land when you want. A custommer usually don't want to land at an airport at 3AM.
   • An airline deveria avoid boarding passengers at 5:30AM, just because if you board at that time, you've waken up at 3:00AM, failed to take a breakfast, almost forgot your papers, arrives at the airport hungry and angry...
   • jet lag : If you take off from San Francisco at 09:00PM, going to New York, you'll land at 06:25AM. It's okay ! But if you take off at 05:15PM, you'll land at 03:40AM. You haven't slept well.. Perhaps at NY, it's okay to land so early in the morning, but try this in Charles de Gaulle and you'll see how boring (or expensive) it will be. Jet lag is an important factor for international flights. When the airline can't have the slots to land at a proper time at the destination airport, it should think of alternate flights to do, or negociate the opening of an intermediate destination of interest before operating the projected flight, this creating legs like "hub-alternal-hub-destination-hub" or "hub-alternate1-alternate2-destination-hub" or "hub-alternate-destination-alternate-hub". Airlines like Air Madagascar does this. Landing at an airport late in the night is another reason of nightstops. This is common with regional jets (737-ERJ-A32x...)

5. Aircraft operated by several airlines.

   • Sometimes, you'll have aircraft operated by the owner some days in a week, and by another operator the remaining time. Take for example InterAir of South Africa. The aircraft is based at Johannesburg, and operates InterAir flights from and to JNB. But on Wednesday and Friday morning, the aircraft takes off from JNB as an InterAir flight, to Cotonou, and from there, it operates for AeroBenin to several destination in central Africa. The aircraft returns from Cotonou the Thursday and Saturday afternoon as regular InterAir flight. So don't be surprised to find a flight from JNB to Cotonou Wed and Fri morning, and their returning legs on Thu and Sat afternoon. That's common in Africa, South America and Asia. Many airlines like Blue Panorama, Lot Polish, EgyptAir (...) lend some of their aircraft to smaller airlines in the world on a week basis like this. The smaller airlines operate them on routes like "hub-A-B-C-B-A-hub" to use the aircraft as long as possible in the few hours/days.

So, those are only hints to understand how routes as selected and arranged. To enumerate all possible schemes, even a book won't suffice.

It is possible to have much more aircraft than routes (when you have enough demand) or the inverse. An aircraft on the ground is an aircraft that costs money for nothing; so "hub-X-hub" flights are rarely the best way to operate an aircraft. Key words are "demand", "costs", "profit". To get factual datas to calculate those, you'll need parameters like aircraft performance, airframe availability, route statistics, crew/maintainance possibilities, security/safety, regulations/laws/authorizations... then compare and negociate the opening of the most profitable route with the less risks with the appropriate aircraft.

Side note : I'm no expert. I'm just an aviation enthusiast, mostly airliners of the 1960-today era.

Depends, sometimes the schedule is tight so the aircraft do several trips without going to the hub. Some companies have designated aircrafts to one specific route (ex: A > B > A > B > A > B and so on). In my experience everything depends on company schedule, aircraft availability, crews avaliable, aircraft inspections, airport restrictions, etc

That'd depend highly on the aircraft and route network.
A 737 doing one hour hops might make 10-15 such in a day's job, a 747 flying a 14 hour leg won't make it back home the same day.
And then there's anything in between.
And whether that 737 flies the same legs over and over again or visits multiple destinations depends entirely on the owner's scheduling department, there's no hard and set rule (maybe it's a -900 which is needed between Amsterdam and London during morning rush traffic, then flies a longer flight to Tel Aviv in the afternoon, and in the evening hops to Berlin and then to Paris before being towed to a hangar for overnight cleaning and minor maintenance).