ricfly744 - the answers given mostly relate to 737 type, as already stated by some contributors.
Back to your original non-type specific question:
MEL applies before dispatch, once dispatched it becomes reference material. For instance if a failure occurs, nothing stops you and thus you may want to check your MEL after having actioned QRH and NNC to ensure that operationally you can continue having arrived at your (nearest suitable) destination or if an en-route technical stop may be more prudent to avoid a lengthy delay due e.g. no engineering cover. Any restrictions in MEL apply before despatch.
Cumulative failures are not accounted for in neither MEL nor QRH, this is where you need to start thinking of how the systems have an effect on each other. It may well be that two independent small failures could have huge consequences but individually treated as non-important.
Q1 can only be answered knowing what type you refer to, as losing a system may downgrade you to a single source of either electric/hydraulic/air is a backup system is no longer working in a twin aircraft, but may not have a huge impact on a quad. In the case of a twin with single course of X you may now have to declare a Mayday, whereas had a backup system still been in place, a PanPan may have sufficed.
Q2 ETOPS (180 or other) has selected entry requirements, thus once in flight prior to entering ETOPS sector you must have e.g. 3 sources of electric, thus on e.g. a 737 which does not have the benefit of a RAT, the APU must be running. This in turn requires a high altitude APU start, or leaving the APU running after engine start until after your ETOPS sector has been completed. Should the APU fail once inside the ETOPS sector, you still have two sources remanning, thus continue not declare negative ETOPS which has further implication such as leaving the ETOPS track if flying in NAT. This APU issue is not the case for e.g. a 767 which has a RAT which when deployed can provide electrical power as a second system.
Q3 The CAT3 briefing requires you to know that all required items for an approach are working, thus are independent of MEL. However, if failure had happened prior dispatch you would've been downgraded to CAT2 or 1 as appropriate.
Q4 FL limitation on 737 is version dependent, NG is better equipped then Classics or older versions, so before an answer can be given to your question here, as some have done, the type reference is required.