Emerging from the sandbox is indeed gradual, you don’t pop out one day. I’ve seen this effect across multiple industries and both old and new sites. Yes, the sandbox hits old sites too (just not as long). For a new site expect 18-24 months for moderately competitive keywords.
As for the very competitive keywords you are going for, without a ton of work, don’t ever expect to rank on the first page of Google. As I suggested before, start looking for some longer tail terms and optimizing for those, the past work will help you rank faster for longer tail keywords.
No, it’s not a good thing to be outranked for your domain name, but if the domain has general words in it (ie: online-gambling-poker-casinos.com), then it may not be a big deal. I just made that domain up, don’t know if it even exists.