- Do companies still offer any kind of learn on the job or entry level positions for people learning to program?
If by that you mean with minimal programming experience/knowledge then, from my experience, not really (for game dev studios). Even as an intern, I was expected to be able to program, implement features etc., but it will largely depend on the company too. Doesn't mean I didn't learn a lot, but programming experience was a must.
- Is it good to try and learn multiple programming languages, or better to try and just focus on one?
Slightly opinion based, but I'll give my own take on it. I'd say focus on one language to start off, so that you actually learn, and eventually know, how to program and have a good take on how to program adequate solutions for a problem. More concisely, as a game programmer, you want to have made at least one game using that one language. I think other languages should not be considered too much before getting to that point.
Not the best of examples, but I know from my colleague students that once they started on C++, the majority was not at said level and thus ended up writing some, uhh, 'special' code.
- Is there a good place to start, skills to learn, if choosing to use professional development?
As for starting out, there's lots of ways to get started. Pick a programming language if you haven't done so yet (preferably use some randomizer to avoid possible forum wars), find some tutorials, try to make some small programs and try to end up creating some game in it. There's some great articles on here as well that can help 'guide you into the games industry'. I personally don't see much benefit in knowing a bunch of programming languages before making sure you can actually program and implement features to a pretty good extend.
Obviously, it's great to be a good programmer, but (verbal) communication is, arguably, even more important. Especially if you are looking to be a lead, this is essential. Along with that, the ability to handle feedback well a need if you ask me, super useful for during code reviews, although that is kind of part of communication as a whole already.
If possible, you could try going for an internship during your studies if you feel you lack professional experience and/or in the above skills. As I mentioned, it has benefited me greatly.
- Do most companies do a lot of remote work, or do they use a physical office?
At both studios of my internships, they rarely did work remotely, mostly due to the product they were working on being NDA bound.
On a side-note, how come you chose 'lead' specifically?