HI There
To answer your first question, with a couple of exceptions, despatching is carried out by the handling agents. At MAN these are Servisair, Swissport (although do check carefully, rumour is that they are all to enjoy a pay cut soon), Manchester Handling, Aviance and Ringway Handling. In my experience, as my brother in law is a despatcher, I would say that either Manchester Handling or Aviance are your best bet. Although not a good time to look at this industry, winter and all that, you might try and apply just into the New Year when these companies have a eye on the summer again.
As for the second question. These items are under the gusies of Airfield Operations. The unit is responsible for Runway inspections, Marshalling, follow me, bird scaring, taxi way monitoring, and safety enforcement to name but a few. As a rule, as this is a division of MA Plc, jobs are advertised internally first. Again as the rule of thumb goes, they like you to have a clean airside driving record, for at least a couple of years, and a thorough understanding of CAP 642 (Airfield Safety, available as a PDF from the CAA website). That said, MAN recently held a recruitment drive for about 6 jobs I believe, by the time I found out about it, the Airport Job Centre had run out of application forms, and they had had 300 + applicants for 6 jobs, so the competition is VERY tight.
My advice, try one of the handling agents for a despatch job. This will get you onto the airfield, as ops jobs do not come along very often, and when they do, the guys on the airfield find out about them first, Joe Public has very little chance at all!