As for the playoff format, it only needs to make the playoff odds equal between the conferences to make the uneven number of teams and odd number of conferences a non-issue:
There are 2 options to achieve this: automatic playoff spots to the Pacific top 3, and Central/Eastern top 4, plus the next 5 best teams by record (call these "Wild Cards" I guess?), not exceeding 7 teams from any one Conference. This way, the number of teams from each Conference that make the playoffs averages out equally: 3-7 (avg 5)/10 (Pacific) and 4-7 (avg 5.5)/11 (Central/Eastern).
Similarly, another option (simpler, but less fair) would be Pacific top 4, Central/Eastern top 5, plus just 2 Wild Cards, resulting in 4-6 (avg 5)/10, and 5-7 (avg 6)/11, giving a tiny statistical advantage to the Central/Eastern Conferences.
So, the first round will always be some form of 4/4/6 split with no crossovers, or 2 conferences will have an odd number of teams. You can just have a single crossover between those conferences, pairing best-vs-worst in the combined group until you get the one crossover, then within-conference matchups for any remaining teams. However, in the case of a Pacific/Eastern conferences with odd numbers scenario, I would instead have one team from the Central cross over with each of the other 2, to minimize time zone travel.
On a similar note, since the Central Conference is made up of Central and Eastern time zone teams, and split 6/5 no less, I would have a rule that no Eastern team in the Central conference is eligible to cross over with Pacific teams, and no Central team is eligible to cross over with Eastern conference teams. This way, teams like DET/CBJ/FLA do not have to play Pacific teams in the 1st or 2nd round, unless there are only Eastern teams left in the Central, which is unlikely with the 6/5 split.
With that all said, here is what the playoffs this year would look like in this format (obviously the results would be different with a different schedule, but for the sake of an example):
Pacific: CGY/VGK | SJ/STL
Central: TB/DAL | NSH/CBJ | WPG/CAR
Eastern: BOS/MTL | WSH-PIT | NYI-TOR
In the second round, if SJ wins, you just have 1 crossover between Central/Eastern, and if STL wins, you have a Pacific-Central and Central-Eastern crossover, likely dictated by which time zone each winner of the Central matchups are. The Central team playing CGY/VGK would not automatically be STL.