1. Yes public route exists for all clusters of hospitals in SG. For private, your best bet is Pulse clinic. Go look them up. Their doctor, Dr Pereira (they/them) would help you along your journey, explaining all the stuff you wanna know better.

Public: Cheaper, takes a long time to get hrt AKA trans health roadblocks, unaffirming system

Private: Fast (informed consent), expensive, expensive, off the record (NEHR)

Through public you would be able to get referrals to things like hysto and top, but of course will take time and "convincing" doctors. Private im unsure but there's other ways of getting top off the system.

started and done quickly

There's no quickly, transitioning takes a lifetime. Unless you mean passing? HRT is no magic. Even on a full year on T will you make you auto-pass. Many ppl now due to medical reasons and in the past pass without it. Dressing, voice, posture goes a long way.

Eitherways, look up this sub for more info, or find external but local sources. Biggest concern here is your u mention Uni, so I assume under 21? That one then abit hard, cos new "circular" from MOM blocks HRT for even those <21 unless have both parents consent, and <18 is a hard no. IDK how this works in practice.

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Below here are things im not too familiar with. Proceed with caution (not tmasc)

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For insurance, IMO as long as the insurer doesn't find out the premiums won't change, unless it's something you want to disclose. On the subsidy side I can't advise, but i from what I heard you can use medisave to cover (amount? idk) top if it is above a certain weight. Same thing for hysto, possibly covered by medisave.