This may sound like a silly question, but does the field perimeter go on top of the mats or go on the side of them?

The field tiles are inside the perimeter in all our matches here.

Does that mean that there will be a small gap between the non-tab parts of the mat and the field perimeter?

You are supposed to cut the tabs on the edge of the mats off. Full tabs on the old mats, half the tab on the new mats. This way the perimeter should not be on top of the tiles, but they should go up to the edge. Keep in mind though that there can be a tolerance of +/- 1".

I have tried cutting off half of the tabs on the new mats, but the mats are too wide for the perimeter; should I cut more?

If that’s the case, then you’ll have to cut more off until they fit. I’m not too surprised that tiles may fit a bit differently and it’s not really a problem, this type of product is usually used where exact sizing isn’t very important, and if each tile is even 1/8" off, that’s 3/4" over the full length which would account for a tab width.

Yes, enough of a gap between the tile and the wall so you can’t lose a wheel or slip down there. A 1/8" to 1/4" on each side is good enough and you will not go crazy adjusting the field walls into squareness.