It is, not period though! If you say repeat after me "wij zwemmen" and you say "we zwemmen" it's simply incorrect, period. Just as "wij zingen" would be. You should repeat exactly what was said, not make up alternatives.
In just a written exercise like I mentioned at the beginning, both indeed are correct in this sentence (since there is no context that specifically requires wij)
I think about the same as in english.
We use auxiliary verbs just like english for other tenses. So have walked is hebben gelopen. So unlike languages like Latin the verb doesn't get tonnes of inflections. (Where something like I might have would have had walked would just be walk with a long suffix) (sorry too tired to look up an actual example)
Many. There are conjugations in all tenses. Check this page for the verb"to be", for example: https://www.vertalen.nu/vervoeging?vervoeg=zijn&taal=nl