Out of these two algorithms. Is there always an input where A is faster then B? (Big theta notation)

Θ(n log n) means the algorithm is O(n log n) and Ω(n log n), and is slower than Θ(n) so your reasoning is correct.

But there's another layer: Landau notation denotes asymptotic bounds on functions. Saying "worst case" restricts it to those functions which result in the highest run time. For example take binary search: we get a best-case runtime asymptotic of Θ(1) and a worst-case asymptotic of Θ(log n).

To answer your question about the difference between "worst-case Θ(n log n)" and "run-time O(n log n)"*, the first one gives us more information: it tells us it runs about as fast as n log n, where the second tells us it runs at most n log n.

_{*Technically the second one doesn't tell us for what 'run-time' configuration it is asymptotically approaching, but in that case generally most people mean 'in the worst case this algorithm runs in O(n log n)' when they don't say anything.}