Can an LL(k) parser parse any grammar without left recursion or common prefixes?
A 'top down parser' is usually understood to be a hand-written $LL(1)$ parser, or $LL(k)$ for some low $k$, which would explain the use of this terminology. These two concepts are therefore essentially the same.

I'm unsure what you mean exactly by 'common prefixes' - I guess you are talking about common lookaheads. If you ensure that you can always uniquely distinguish which production you need to expand by looking at the next letter, then your language is $LL(1)$ (and also contains no left recursion). Doing that usually means you change your grammar structure a bit so your parser is never confused by 'k letter prefixes' of the part of the input that you haven't looked at yet.

If you are really talking about common prefixes in the language-theoretic sense, then all your words would have to start with a unique letter, which means your language is finite and therefore $LL(1)$.