What is the ruling on prayer after entering a swimming pool and sufficing with ablution without performing ghusl? And what should a worshipper do if the congregational prayer begins while he is performing تحية المسجد (two rak'ahs upon entering the mosque), whether he is in the first or second rak'ah?

Swimming does not nullify ablution (wudu') nor necessitate ritual bathing (ghusl (definition)). Therefore, if you are in a state of purity, prayer is permissible after swimming without performing ablution or ghusl. However, if your ablution has been nullified, you must perform ablution for prayer.

Regarding the ruling on interrupting a supererogatory (nafila) prayer when the congregational prayer (salat) is being established: if you are in the second unit (rak'ah), complete it swiftly. If you are in the first unit, interrupt it, because whoever catches one unit before the congregational prayer is established has caught the prayer.