If a program wants you gone, they will usually offer to let you out of your contract and some sort of letter with what you got credit for in that program so that the new one can figure out where you fall in their program. If they want to add any editorial to that, it definitely could happen. Leaving quickly could avoid termination, which they often make clear and is truly better for your record. They could also negotiate to have minimal to no adverse reports on what is released to other facilities for leaving quickly. With regard to credit given at your new program, assume the worst, no credit, and be pleasantly surprised if they offer you 6 months to 1 year of credit.
Every app for everything related to medicine will have some form of those "have you ever" questions on them. Answer them honestly, but you may want to consult an attorney to see what language you could use for the excuse. If you had a 2 month gap, you definitely could have been reviewing literature relevant to your field, or working at a job to pay the rent, so put that. As long as the first clerk sees an issue and a feasible explanation has been provided, you'll likely pass to the next round of scrutiny.
If not, your app gets escalated to an administrator to review and, yes, your time to reach credentialing may be extended and you'll have to push out your start date. It may take an extra week or two, but that should be it. Anything criminal is a whole other story. Make sure your program is aware of any possible delays, not necessarily why if they ask or say nothing and blame red tape. If they want you, they'll wait.
Resign with class, shaking hands and giving your support staff hugs if they're open to it and make sure that your contact information is up to date. They won't call, but it's a nice gesture. Giving 30 days notice is usually the goal for professionals, but you could play dumb and give two weeks and blame it on a rookie mistake if you're ever questioned about it. Still turn in the letter no matter what, since the people at the program may change and the next group can see that you properly resigned.
I take the above from others that have commiserated with myself about the quality of their programs, and bad outcomes have come to those not prepared to handle the circumstances. Residents talk, ask around.