Contraction Faction, What's Your Action?

Contraction puzzlers:

1. If I'm is a contraction for I am, why does it sound wrong to say:

I think, therefore I'm.

2. If you've is a contraction for you have, why does it sound wrong to say:

Give me all the money you've.

3. If he'd is a contraction for he would, why does it sound wrong to say:

He'd, but I wouldn't.

4. Why does it sound acceptable to me (West Coast American English speaker) to say:

There's more carrots in the refrigerator.

even though the uncontracted form is clearly ungrammatical:

There is more carrots in the refrigerator.

Obviously, this should be There are more carrots . . . But there're is a very awkward contraction. So there's sounds better even though it is wrong?