Will an outlet adapter installed in a motion sensor light be constantly hot?

If I understand correctly, you'd like to screw a plug adapter into one of the bulb sockets on your flood light. If so, here are the caveats that come to mind:

• The outlet will only be live when the adjacent bulb is lit. It would use the same electrical contacts that the bulb would.
• There will be a current limitation that you could easily violate by plugging other devices into the light socket. The socket and its connected wiring are designed apenas for the current load of light bulbs up to the maximum listed wattage. In the case of a 75 watt flood lamp, that's less than one amp. The wiring may be fine up to say 5 amps, but we can't know for sure, and you open yourself up to smoky outcomes if someone else comes along and plugs a table saw into it.

In general, it's not a good idea. Used with caution, it could work for a short run of holiday lights, for example, without issue.

Yes, that will be fine. Those 2-light sensors are packaged units that get supplied always-hot. You can tap that.

You should turn off the breaker before doing any work.

Most flood lights have a interior wall switch that regulates when they work or not.

For wiring an external weather rated receptacle that is GFCI protected and accessible, you probably should put the GFCI in the panel as a breaker ( not always a DIY job ). Per 2017 NEC the GFCI should be accessible.

When wiring the the GFCI protected receptacle to the light, you should see a black, red, white, and ground. The black wire would be pig-tailed to supply the permanent power.