Does the ops manual stipulate that a flight once airborne must hold an alternate if the weather at destination is below alternate minima?
Yes. Can't speak for every airline but that is the rules in Australia. Now you have options within that like using a SLAM plus the TTF for arrival, but if it changes to Fog at 100m then you will either have to have a alternate, or enough holding for the FM period commencement. I believe that is where one of the Qantas incidents got caught. The TTF had it clearing up but it didn't.
Don't know why Australia has a ceiling requirement.
The only solution to all this would be to have European styled rules with compulsory alternates at planning stage and CAT IIIB at all capital cities. That way you could carry on and have a go at the Autoland.
However in Australia that would a very expensive solution and never get passed by the airlines which is why we have the no alternate rules. Some CAT III gear would take the stress out though.