Can a 2nd Year PE Analyst Lateral to Another PE Fund Off-Cycle?

Hi WSO! I am coming up on my 2nd year as a PE analyst at a well-established MF ($100B+ in AUM). I am currently in a smaller US city and have wanted to relocate to New York or San Francisco since I started the job. I'd like to stay in a similar role but my firm does not have a presence in NY. Below are my questions regarding the off-cycle process:

• How challenging would it be to lateral to Associate roles at major mega-funds like Blackstone, Silverlake, Thoma, etc. right now? I noticed BX has a 2025 Associate role open on LinkedIn right now. Should I just start reaching out to folks on the team I'd be interested in?
• How common are immediate-start openings? Isn't that a bit of a sign that the firm has churned out most of its junior team due to cultural issues or workload? Ideally, I'd start after bonus hits in January
• How important is networking? How common is it for off-cycle opportunities to exist at established firms? Do I have to go through headhunters?
• How likely is it that a headhunter would withhold opportunities from someone with my background? I spoke with a couple this past week, and both said they would share my profile with the opportunities I found interesting. How long should I expect to hear back about interviews? Will they let me know if I happen to be passed up for any opportunities?
• Any general recommendations on interview prep? I haven't interviewed since I applied for my summer analyst position, which I accepted a FT return offer for. I can confidently model and talk about my deal experience, but I'm not sure how polished I'd be if someone asked me off the cuff to walk through $10 changes in depreciation through the 3 financial statements....