Is it true that after taking the SAT three times officially, your scores do not count afterwards?

It's not that it doesn't count, but colleges look at it unfavorably. I would say take it once just to get a feel for it, then practice on your own time with study guides and/or test prep classes before you take it a second time. Reserve the third for if you're unsatisfied with your second score.

Nope, there's no limit to the number of times you can take it. However, from a college's standpoint, the fewer times you have to take it, the better.