Does a 403(B) require an annual 5500 to be completed and submitted to the IRS, like a 401(k)? Does discrimination testing need to take place?

Yes to both, if the plan is covered by ERISA. I'm really simplifying this, but at a high level, a plan is probably covered by ERISA if it has employer dollars in it and/or the employer exerts some sort of control over the plan, like approving/reviewing distributions, selecting only a specific investment provider or funds for the plan, etc.

A couple of additional comments: a 403(B) does not require either discrimination testing (except for the universal availability rule for salary reduction contributions) or Forms 5500 if it is a governmental or church plan. If it is not, then JWBrown has provided a summary of the ERISA coverage rules. Note, however, that the 403(B) nondiscrimination rules for salary reduction contributions are much less onerous than the comparable 401(k) rules, as they require only universal availability, not ADP testing of actual contributions.

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Employee benefits legal resource site

I think you have to do ACP testing if there's a match.