Yes, we do say that in English!

To me, it would only mean that in a very definite context. If someone just said to me, "He's a strong teacher", I don't think I'd assume that meaning. However, this may be because I've been teaching ESL/EFL for so long and have heard 'strong teacher' used incorrectly to mean 'strict teacher' so often. That might be colouring my interpretation. For me, 'he has strong teaching skills' is much clearer.

That would sound a bit unusual to my ear. If I thought someone was a great teach, I would state they were an "excellent" teacher.

Yes, we say that, use 'strong' in that sort of context. It's fine.

In most cases people just say "good teacher" or "really good teacher" if you're talking about their teaching skills. In some cases you can say "knowledgeable teacher" but usually it's when they know how to explain the topic deeper than you would expect them to.