Can I establish a $2000 Roth IRA for 2001 before April 15, 2002 and then contribute $3000 for 2002 shortly thereafter (in May)? Thanks!

Absolutely.

When you contribute the $2,000 for 2001, make sure that the custodian understands that it is a contribution for the calendar year 2001. Otherwise, they'll likely code it as a 2002 contribution.

You can actually do the full 5k anytime before April 15, even on the same day. But, as noted above: be sure the custodian understands the designations. Best way to do this is (1) use two separate checks and put the contribution year on the check, and (2) double check your monthly statement to make sure the funds were deposited and properly designated. Year designation mistakes are pretty common in the overlap period.