Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2 $squares, with sides $1,2\ldots n^2$ (n odd). Which seems like it would work with ANY odd $n$. It's so simple, surely it's well-known, but I haven't seen it in my (brief) web travels. I have a page with pics here of the $n=7,9,11$ versions, and description of how to construct them.
It seems the same method could square the plane, well, fill greater than any specified area, no matter how huge, at least. Which is what 'infinite' means, practically, isn't it? Anyway, it seems, if somehow not known (which it must be, surely - if anyone has links etc to where it's discussed I would be very grateful) then it's another way of squaring the plane.
Q2. Each of these arrangements can be extended, but the ones I've tried (5 or 6) have a little gap to the south-east, (i.e. where the squares don't fit neatly together) but otherwise can be extended forever. Is there an n for which there is no gap? There are more than 1 of each sized square in this arrangement, but still, it would be a nice tessellation with integer squares. Here's a picture of the 7x7 version extended, and detail of the centre.