Is this proper English...

1) Tom, please make sure you keep Joe and I (me) informed.

The proper English is 'me', because it's the object, along with Joe, of the verb 'keep (informed)'.

2) It is obvious that he trusts you more than I (me).

If you mean he trusts you more than he trusts me, again, 'me' is the correct form, because, along with you, it's the object of the verb 'trust'. With the 'I', the sentence could be read as: It is obvious that he trusts you more than I trust you.

There's a tendency for people to use 'I' when they should use 'me' when there's a double object like Joe and me, you and me, above. It's usually explained as 'hypercorrection': people know they often use 'me' when they should use 'I' (for example, "It's me", which, technically, should be "It's I" - very stilted!) so they respond by using 'I' in places that they shouldn't.

If you're unsure about which form is needed, try taking out the other object and see how it reads. Would you say: Tom, please make sure you keep I informed?

For more information, see Are we agreed?