It looks like it is possible to rename trade gates.
Is this allowed?

The trade gate that I have is showing orange, as in it doesn't have any orders this turn.
How can I turn it green, or does it not matter?

My trade gate is saying the following when I click on it:

Cannot activate contruct

Is this supposed to happen?
When would my goods start flowing to my homeworld?
Do I need a trade gate in my home world system?

Trade gate rename is allowed. No problem there, except maybe your own fleet maintenance.

You do not need a trade gate in your homeworld.

You CAN make the trade gate turn green by constructing armour on it. But you do not need to. The server processes the economy every 6 hours, and goods automatically flow through the gate unless it is contested (more than 1 race has ships within 5 hexes).

When it says you cannot activate the trade gate, it should really say something like 'Trade gates need no orders'.

Trade gates do not count towards the normal solar turn flow - so if you are in a system and there is an enemy gate you should still be able to send in lots of turns one after another. (Note: I think I tested that....)

How do i repair a trade gate?

You cannot!

I'm not sure what to do about that at the moment. In one way it's quite good. In another it's annoying.

You can self destruct it, and then seed another next turn. Which seems a decent work around to a situation I can't fix (for now).