About second question ・会う：Llike "会議", "集会" and so on, basically this kanji "会" has implication "many people gather". But we normally use it regardless of the number of person. ・逢う：Like other answerers wrote. And this has implication "finally", "to meet at one point" and "from both directions". I hope it will help you :D

頑張ろう Work hard to achieve a target that you might have. Your friend doesn't know your target exactly. お互いに =　each other 会う　meet 逢う　 ①Like Ken wrote. Used when one want to express one's emotion、such as meeting lover. ②meet but god made us find each other , fateful encounter. 　 There are some more ! 合う　fit 遭う　meet coincidentally

I agree with Nelly's first answer. I usually say that when you and I both have a bad or hard situation. Fore example, if you and I seek a new job but we cannot get any job for a while, then I say お互い、大変だけど頑張ろう。 About second question, we normally use 会う. But if I meet someone like special person for example my girl friend or someone I respect. Then I write 私はこれから恋人に逢いに行く I'm going to see my girl friend from now. And this sentence includes my passion for my girl friend as well.

Can't answer your second question, but the お互い in the first sentence means something like "both", therefore お互い頑張ろう means something like "Let's both try our best".