I saw the word Працювати and in past tense it was працювали, why isn't -вати removed completely not just -ти. (There was no information below the lessons). Also what would "for" be when used with verbs, example: for doing this...?
Why do you think that "ва" should be removed? "Ти" is the infinitive ending, thus it's removed. "Ва" is just a part of a verb (there are a lot of verbs without it: летіти, стрибати, говорити). You can see some past examples here: https://www.duolingo.com/comment/9175226
I didn't really understand your question about "for"
In present tense 1 it said any verb with the infinite -вати is declined by taking -вати off. And for "for" I mean like if I said "thanks for doing this", what would "for" translate to? It even says to drop it here: http://www.ukrainianlanguage.org.uk/read/unit10/page10-3.htm