Safely wiring a 120v smoke detector externally
Is standard gauge wire fine as long as it's rated for 120VAC?

You can use any wire with sufficient insulation if properly fused. For this refer to your local electrical code. The factors are wire surface area (awg) and length, those will give you a maximum current.
Typically for device wiring 3x0.75mm2 or 3x1mm2 is used. (17/19 awg)
For smoke alarms a 5x20mm 1A Slow fuse should be sufficient, as found in appliances.

How should I splice the connection to the thin wires on the smoke alarm, wire nuts?

Today you can buy better splices than simple twist on wire nuts. Like Wago 221 terminals that can clip on to anything between 0.14mm2 to 4mm2 stranded or solid.
Obviously above does not provide you with physical strain relief. You'd have to buy smoke alarms that feature this (industrial) or create your own.