The ability to filter the trade gates from the ship list on the right hand side of the screen is great. The drop down button is partially hidden though.

Also, it looks like it is possible to set the trade gate to jump to the galactic map. Is this allowed?

Trade gates jumping should not be allowed - a bug.

The filter being partially hidden? Hm.
What OS are you using, and screen res? It's fine on my PC's.

I am using Windows XP with the Firefox browser
display is 1024 by 768

oh.

I have seen a trade gate that gets a warning
that it must be 20 hexes away from a planet
to jump out of the system.

This is not needed?

It is possible to target your own trade gates.
A half blank drone screen appears but you cannot fire the drone.
I think you should not be able to see this.

You are probably right about that... It's on the list.