It's beat in both senses, 'hit' and 'beat' (as in 'win against/versus', obviously the one meant here). It also means strike as in klockan slog två, 'the clock struck two'.
Can it also mean something like "surprise" or "catch off-guard"? At least in a sentence like this one?
Not in a sentence like this one, but there's an expression slå med häpnad which means 'surprise' or 'stun'.
slain is the past participle of slay which means 'slå ihjäl' or 'dräpa' in Swedish.
Nope. Sluta is from Low German "Slûten" meaning to close or finish. Slå is from Icelandic "slá" and/or German "schlagen", meaning to hit, and is related to the English "slay".
It's literally impossible to hit her. She has an invisible force field around her at all times.