A small distinction, it's FRCPath (Fellow of the Royal College of Pathologists) that's useful, not FRCP (Fellow of the Royal College of Physicians) for these jobs.
1) You would almost certainly need FRCPath and clinical scientist registration but you don't need to go through the HSST route to get them. The HSST structures the pathway and gets a training budget etc.
2) A sponsor is someone to sign off that you're ready to go for the exam, it's usually someone with full FRCPath accreditation. By a support network I mean having people around you who can help to answer questions and give you advice on how to do the exams, I'd have found it impossible to get through without them.
3) For most disciplines FRCPath is 2-3 exams, each one gives you 4 goes (per person per life, although you can try to ask for extra attempts if needed). You only need to pass the STP/equivalent routes (I think it's the AHCS certificate you're referring to) to be a clinical scientist, you then normally need to pass the first bit of FRCPath to get to the intermediate jobs and full FRCPath along with everything previously to go for consultant jobs. A doctorate is useful for consultant jobs but usually they also say "equivalent experience" is acceptable.
Again, this is my experience in biochemistry and a few related disciplines, I can't speak for genomics specifically.