> Duration of immunity? I've heard a minimum of 3-4 months reported, possibly longer, but do we know more now?
The only way to know for sure is to wait and see. But patients vaccinated for SARS in ~2004 still show immune responses now and people who received the various SARS-COV-2 trial vaccines in April of this year still show immunity. There's also studies showing many people who recovered from COVID on their own have demonstrated 9 months (and counting) of immunity. The truth is that the immune system is super complex and there isn't a single simple blood test they can do to tell if a person will resist a virus or not, so a lot of the conflicting numbers you will have seen depend on what exactly was tested and the media reporting has been pretty poor at contextualizing those study results.
So there's reason to hope vaccine-derived immunity could be long lasting (say 1-10+ years), but we'll just have to see. But talk of 3-4 months of immunity seems to be media hype.
> Am I right that we don't yet know if they reduce transmission or simply prevent the disease?
Correct, we don't know yet. Pfizer unfortunately only tested people with symptoms for COVID in their trial, so we have no idea if it reduced asymptomatic cases or reduced transmission. Oxford did test thousands of people weekly in the UK to find asymptomatic cases and they have said they have "evidence" their vaccine reduces transmission, but they haven't published their peer-reviewed Phase 3 results yet so we don't know by how much. The scientific consensus seems to be that all the vaccines will probably reduce transmission by some amount, but by how much is unknown. But it stands to reason that the quicker someone clears the virus from their body, the less time they will be shedding infectious levels of virus.
> Short of eradication, if immunity is not permanent, what options do we have apart from a perpetual vaccination program or just accepting an increased rate of mortality going forward?
Luckily, this probably won't be the case. In any case, viruses tend to mutate and become less virulent over time since killing your host is a bad way to transmit yourself. So the general guess is that the virus will eventually mutate to be less lethal, similar to the how the 1918 flu virus still circulates but doesn't cause that level of mortality anymore.