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
Thank for, pay for, vote for, fight for - with these for is translated as за.
-ВА- is dropped in present tense conjugation (there are exceptions!) and it stays in past tense.
Your example (for doing this) uses gerund, you can't directly translate it to Ukrainian. You need to use a couple of pronouns or a noun: дякую вам за те, що зробили це, дякую вам за допомогу/ за те, що допомогли (thank you for your help/for helping me)