Ramin Rahamim said:

1. Does AM26LV32 support hot insertion?

This device is essentially just a receiver so the only thing you need to concern yourself with is the transient that it may see when you initially make a connection. As long as the A/B pins don't see a voltage outside of its abs max ratings it should be fine. The abs max for this device is +/-12V which should be wide enough in most scenarios. 

To make it more 'hot swap' friendly I would recommend putting a RC circuit on the G/G# pins to enable the device after a few miliseconds (I believe hot insertion events last 100s of microseconds so you just need to pick a value larger than the hot insertion time).

The other thing important to hot insertion is how the connection is made. Ideally you should make contact with GND first then Vcc then the signal lines. The app note below walks through hot swap tips.

https://www.ti.com/lit/an/slla107/slla107.pdf?ts=1752609262658&ref_url=https%253A%252F%252Fwww.google.com%252F

Ramin Rahamim said:

2. What happens is signals are connected but there is not power to the device?

It won't break but you may end up seeing some leakage current. Some of the leakage may make its way to the Vcc rail. 

Ramin Rahamim said:

3. The same questions for AM26LV31.

Same comments as above but this device is a driver, you it's more important that you ensure the G/G# pins hold the device in a disabled state until the device is fully on. 

In all systems you will see some sort of narrow glitch on the bus when the hot insertion event occurs due to the parasitic caps on the secondary boards charging up when they are inserted. 

-Bobby