which versions of sqlite may best suite for python 2.6.2?
If your Python distribution already comes with a copy of sqlite (such as the Windows distribution, or Debian), this is the version you should use.
If you compile sqlite yourself, you should use the version that is recommended by the sqlite authors (currently 3.6.15).
I'm using 3.4.0 out of inertia (it's what came with the Python 2.* versions I'm using) but there's no real reason (save powerful inertia;-) to avoid upgrading to 3.4.2, which fixes a couple of bugs that could lead to DB corruption and introduces no incompatibilities that I know of. (If you stick with 3.4.0 I'm told the key thing is to avoid VACUUM
as it might mangle your data).
Python 3.1 comes with SQLite 3.6.11 (which is supposed to work with Python 2.* just as well) and I might one day update to that (or probably to the latest, currently 3.6.15, to pick up a slew of minor bug fixes and enhancements) just to make sure I'm using identical releases on either Python 2 or Python 3 -- I've never observed a compatibility problem, but I doubt there has been thorough testing to support reading and writing the same DB from 3.4.0 and 3.6.11 (or any two releases so far apart from each other!-).