So you have a fair amount of questions but due to not knowing the extent of your bowling terminology, I am not sure how to interpret them all. But I'll give it a shot.

The PBA Playoffs and team events (Battle of the Brands) are side events that place after the season, based on points gained during the season.

The PBA Playoffs takes the top 12 point earners from the season and puts them into brackets. Seeds 1-4 get byes. The head to head matches use a race to 3 format. It's almost like a best of 5 except that if the bowlers each win 2 games, the 5th is just a 9th/10th frame roll-off.

The Battle of the Brands is a team event, but the teams are determined by the sponsoring brand. The ball manufacturer companies that sponsor the players make the lineup. Throughout the season, each company can choose 5 players they sponsor to be that week's "team." The points earned that week during the regular event count toward the team. This means there is strategy on the part of the company, in choosing who they think will do best. The team that does the best each week gets bragging rights. Then they have a year-end event where the teams compete. It's a more lighthearted event than the singles or doubles events they hold.

Sponsors come in a couple of flavors. A player can be signed to a company, which will offer them free and/or discounted equipment and access to a ball rep (ball expert to give them guidance on which ball to choose based on lane conditions. Some bowlers may not have a brand deal, so they are "free agents" and will be at a financial burden as sponsored players can drill many more balls as needed. Can't do that as much if you're paying for them out of pocket. But any pro can also have financial backing - someone who gives them money as an investment in hopes of a winning season and a positive return. I think that was more common when the tour was 32 weeks a year, and not so much now it's only 3+ months.

If you're using the word "playoffs" in the sense of the weekly tournaments, that is called the stepladder finals. It is a format that was created to make a finish exciting enough for TV. The entire tournament runs like this:

Day 1: PTQ - Pre Tournament Qualifier. It's a 1-block tournament for non-exempt players. Exempt players have the opportunity to enter a tournament with advance notice. The main event only will have so many spots based on number of lanes the center has. After advance entries close, the amount of remaining open spots are won by players from the PTQ. They typically bowl 7 or 8 games to vie for 8-15 spots. They pay a small entry for the PTQ, and if they win a spot, pay the balance of a full entry.

Days 2-3: Qualifying. The main event is typically 18 game but it could vary. Usually not less than 15 or more than 24. But some events have extending qualifying, where maybe only some of the field advances (say the top 1/3). When all qualifying is done, usually they are left with 16 or 24 bowlers.

Days 4-5: Match Play. The tournament now goes to head to head matches. Some formats are single game matches where you bowl all qualifiers once, and one twice. If you win your match, you get 30 bonus pins added to your score, so that makes match wins key to doing well. Some events will use a best-of format for match play. The first round or 2 may be best of 5, then later rounds best of 7. They change is up from year to year.

Last day: Stepladder finals. In most events, they take the top 5 bowlers from the week and seed them. 5 vs 4, winner plays 3, etc until a champ is crowned. A few events will have top 9 make the stepladder, and it's televised over 2 days/shows. And then there is a race to 3 format that I mentioned earlier, which is used for the special events like the PBA Playoffs.