I'm riding a 1972 american eagle road bike frame, and I'm wondering if the position of my rear wheel can't be changed (it's got a few inches of play in the rear drop out, and a few inches from the seat tube). I guess I could remove some chain links but..

1) is this a good idea and,
2) will this have any effect on my ride?

Thanks in advance!

It is pretty much the default practice to position the rear wheel back and forth in order to properly tension the chain.

Sure. I understand that. I'm just wondering if too much space between rear wheel and seat tube makes any difference...and if so, should I think about narrowing the difference.

If I understand your post correctly...

1) Neutral
2) No

If you have horizontal or semi-horizontal dropouts, then you don't have to worry about how far back the wheel is (further away) from the bicycle. If it's too far back, it certainly isn't going to fall out (as they bend back around). I would be more concerned with it being too close to the start of the dropouts. If it really bothers you how far back they are, cut out two links from the chain. As long as you have a tight chain, I wouldn't worry about it. If the chain is long enough where you're as far back in the dropouts as possible, and still have slack, cut it.

In general, the shorter wheelbase will give you quicker handling. A longer wheelbase will give you better stability.
But in real practice, most people won't notice the difference.

It might have a tiny effect on how the bike handles, but it is probably not worth the effort. I don't know that frame, but if it's like other lower end bikes from the 70's, it probably has a very low seat tube angle, which will cause a lot of space between the wheel and the seat tube. There's not much to be done about that.