Regarding the question in the final paragraph, one can't find infinitely many non-collinear points, that's the Erdős–Anning theorem. It's one of those marvellous results where the mathematical review contains the entire proof...
However, it is possible to find any finite number - eg by finding an integer which can be written as the difference of two squares in lots of ways and then drawing the corresponding right-angled triangles on top of each other. It doesn't help at all with your earlier questions though. An interesting follow up question is to find points with no 3 on a line and no 4 on a circle - this is Section D20 of Guy's Unsolved Problems in Number Theory - apparently no 7 point set is known.
Edit: change that 7 to an 8, cf Tony Huynh's answer!
