Team championship pairings Is there some fixed rule how teams for leagues are paired? Is it deterministic once team gets its starting number?
If for example two teams must play in early rounds (e.g. A, B from same club), is it always done by giving them appropriate starting numbers or are some leagues where rounds are done manually? Are there more pairing standards?
Do some leagues have an odd number of teams and would it change something?
A lot of questions with many different answers depending on the type of competition. Leagues can be either Swiss or round robin. I suspect most leagues are round robins but by no means all are.

Perhaps the most interesting league from the viewpoint of pairings is the the UK's 4NCL (originally 4 Nations Chess League). There 4 main divisions with subdivisions and both Swiss and round robin pairings are used according to numbers. If you are genuinely interested then the examples from this league are worth investigating. The rules are here - http://www.4ncl.co.uk/1718_rules.htm


  Is there some fixed rule how teams for leagues are paired?


No. There is no one fixed rule. If there are roughly the same number of rounds as teams then a round robin (all-play-all) pairing system would be used which does have fixed rules. If there are many more teams than rounds then some form of Swiss is used. Sometimes the league might be split into two time-wise with several subdivisions paired with one system (rr or Swiss) for the first half and then rejoined depending on results into separate sections which may again be paired in different ways. Again there are rules but some are fixed and some mutable.

More information on Swiss rules are available on the FIDE website here 

For round robin tournaments Berger tables are used. The only reference I have for these is the FIDE Arbiters Manual These start on page 66.


  Is it deterministic once team gets its starting number?


For round robin, yes. For Swiss, no.


  If for example two teams must play in early rounds (e.g. A, B from
  same club), is it always done by giving them appropriate starting
  numbers or are some leagues where rounds are done manually?


It depends entirely on the circumstances. For a round robin it is easier to do this by assigning start numbers appropriately but it is not essential.


  Do some leagues have an odd number of teams and would it change
  something?


Of course some leagues have an odd number of teams. For round robins, where the Berger tables are used, one team has a zero-point bye each round. For Swiss systems triangular matches are used.

As I said at the beginning the UK 4NCL is interesting and complex with a full range of pairing and worthy of study if you are really interested in this sort of thing. 

You might find Division 3 North of particular interest because it has a relatively large number of teams. The discussion on this (2017-18) season on the English Chess Forum is here . The arbiter's job was complex. This post shows one of the interested parties posting the ingenious pairing rules that were decided on by the arbiter to solve the problems posed.