You can say: "din timpul verii". "Cursul" is synonymous here with "scurgerea" passing, spending, running out. For example, you can say: "cursul timpului" or " cursul apei", although in the second example indicate the direction also.
'I have worked here since the summer' or 'I have been working here since the summer' would be better translations, IMHO (present perfect -- or present perfect continuous -- for ongoing actions that have started in the past).
Not simply better, but the correct translation(s) (with "work" and "live" we can use either). DL's English sentence here is simply ungrammatical (i.e. no native speaker would say it), and I'm really surprised nobody picked up on this before you. So have a lingot. :)