Yes, there is a difference. 1. we'll see each other (most likely this is a person that you already know) 2. I'll meet him or We'll meet him (in a cafe) 3. we'll see him (might be a rockstar on a concert, you can see him, but he can't see you)

Hi. "Мы увидимся с ним" and "Мы встретимся с ним" have a similar meaning. In the vast majority of cases, this means that you have appointed a meeting with someone in advance or are going to meet with this person again. You are going to see a certain person, and this person also wants to meet directly with you. "Не волнуйся, мы увидимся с ним позже." - "Don't worry, we will see him later." "Я договорился с Энди. Мы встретимся с ним у метро." - "I made an appointment with Andy, we'll meet him at the underground station." In the third case (Мы увидим его), you just say that you will see someone (a politician, an athlete, a movie star, etc.). "Завтра в наш город приезжает известный спортсмен. Мы увидим его на стадионе." "Tomorrow a famous sportsman comes to our city and we will see him at the stadium."

No

1. It would be better to have more context (as usual). This sentence can be neutral, not rendering any emotions, but it can also be "emotional'. In the first case, you may talk like this about a colleague who you know you'll meet because you work together. (You can say "Мы увидимся с ним tomorrow and I can ask him about that matter.") In the second, "emotional" meaning, you can use this sentence when talking about someone you're willing to see because you are attached to them or because you haven't seen someone you used to be attached to for a long time (and this someone is attached to you as well). It would be like "I am looking forward to seeing him - ("Наконец-то мы с ним увидимся"). Sentence 3 can aso be used like this, with a little difference - it would mean that you miss the person and he probably doesn't. 2 and 3 have been explained well here.

As abut difference between "увидимся" and "встретимся", the latter may hint on an appointed meeting, the former doesn't, it may be both an appointing meeting or just you come to the same conference/party. But they are interchangeable even in the context of a conference.