Would a rogue with the Minor Magic talent qualify for Arcane Strike?
Nope. Minor magic isn't an arcane spell. It's a spell-like ability, which are not the same things as spells. Thus, you don't meet the prerequisites of Arcane Strike.
Okay. The way the text is phrased made me wonder. The word "cast" is used repeatedly.
Yea, I have a thread with a FAQ request that'll hopefully help resolve that. Basically, that's one of the few placesi n the rules that uses 'cast' to mean 'activate an SLA', and it does cause confusion.
Truth be told, I don't even think SLAs are arcane or divine. They're just...magic, you know? The description of them says nothing about them having a magic type, and the alchemist shows us that there are definitely things that look like arcane spells but aren't arcane (or spells). So it fails on multiple accounts.