First of all, when interviewing with the potential new employer do not mention “all of the problems with your current employer”. Every employer has problems and shortcomings – and so will your new one – no one wants to hear a litany of “what’s wrong at such-and-such company”.
Feel free to mention “time for a change” (or a new challenge, or new goals, whatever) and leave it at that. (That is, unless asked to name some of your “new goals”, in which case you should have one or two on the tip of your tongue, or be outed as a big-talking liar, even though that wouldn’t be expressed overtly by a diplomatic interviewer.)
Second, depending on the position that you’re being hired for, the process can frequently take months, literally, for a full-time permanent staff position. Even in a project-based company such as ours, where semi-permanent professionals are hired to fulfill contract positions of absolute necessity to meet milestone objectives, we often have to start the process months in advance to be certain to complete it by the time of actual need. But that’s often because those hiring decisions cross international lines, and employment visas and immigration issues are part of it. Presumably that won’t apply to you, but still… hiring processes often take longer now than they used to. Even our US permanent staff hiring among nominal American citizens still has to start with a budgeting slot in the year before the hire is made, then a multi-step interview process, etc.
Assuming your hiring process will normally be quicker, say, for an administrative or lower-tier production-type employee, it is perfectly fair to mention to the interviewer that you have “a long-planned family event” (I would not use the term “vacation”, because it cuts across the Calvinist grain of a lot of corporate types) scheduled for February, and that your availability is limited by that one event.
Alternatively, if you have not yet paid for or made firm arrangements for the vacation, I’d suggest simply putting it off, because starting a new job is adventure enough for a lot of people. You won’t have gotten into the long-term grind and stress of the new job, and its novelty should keep you fresh for awhile.
I never worry about those “personal revelation” open-ended questions that interviewers ask, because they’re only about getting you to speak about something to hear how you express yourself, the vocabulary and tone that you use, a revelation of your thought processes. No one actually cares about your biggest flaw, your personal goals and your insights into life itself; what they do care about is “how do you think and speak and marshal an argument”, so work on those things. A good way to do that is to participate more here, and in just this way.