1. Bienvenus is an adjective¹ here (despite being described as substantive by some sources). In this specific construction, les introduces a subset of a group, the ones who are welcome vs the ones who are not, the" welcome ones" vs "the unwelcome ones". Note that there are not a lot of adjectives that can be used both ways like bienvenu. Here are a few that can: on est perdants / on est les perdants, on est premiers / on est les premiers, on est nouveaux / on est les nouveaux, on est seuls / on est les seuls.

2. We feel like being the welcome ones. Here the pattern is se sentir + subject attribute.

3. The verb must grammatically agree with the subject so is singular but the adjective must logically agree with the subject. Here on represent more than one person thus the plural. The same happen in English with the pronoun "you" that requires a plural verb "are²" but can have a singular or plural subject attribute depending of who is behind that "you", e.g.: "you are the man" vs "you are the men". A football chant became famous after the 1998 World cup final: On est les champions, on est les champions… (We are the champions)

4. Se sentir bienvenu is absolute (I feel welcome) while se sentir le bienvenu is being part of a class (I feel being part of the ones being welcome). Many sentences that use être + adj. can use se sentir too, with of course a difference in meaning. Je suis prêt / je me sens prêt (I’m ready/I feel ready), je suis capable / je me sens capable, je suis seul / je me sens seul. Some are less or not idiomatic je me sens sûr, je me sens content.

5. There is no significant difference. On se sent bienvenus can be heard but seems to me less "French" than on se sent les bienvenus. It is possibly an anglicism.

¹ _{Like adjectives and unlike substantives, bienvenu agrees in gender and number with the subject.}
² _{The singular thou art being obsolete.}