So you would recommend signing up at 16, even if my personal odds of dying now are pretty small?

I haven't signed up myself yet, and I'm 28, so it would be suspicious if I recommended this. (My reasons for not doing so are somewhere between financial and "ugh, paperwork.") But I do recommend becoming actively involved in the cryonics specific subset of the transhumanism community and making long-term financial plans with this expense in mind.

What would you estimate as the probability of developing technology that will make cryonics work without a singularity?

I assign perhaps 65% likelihood to that, assuming we rule out long-range as well as short-range singularities, and assuming something like today's best cryonics quality levels. The added risk is because a) the ceiling for non-singularity tech might be lower (it sort of rule out matrioshka brain kind of stuff, if that is relevant), and b) non-singularity tech could take longer to reach high enough levels (even as compared to distant singularities) which increases storage-time associated risks. The latter kind can be mitigated to some degree by increasing storage security -- but it would have to be very close to arbitrarily low to survive for multiple thousands of years.

The likelihood of it working climbs to closer to 100% (or at least towards the chances of curing aging, which I think are in the high 90%'s in the long run, even without a singularity) if the cryostasis technology improves during our lifetimes, as the problems of cooling can then be resolved sooner and with less chance of being inherently unsolvable (for any given technological ceiling).