I've posted a ton on objectives and game setups, as well as gametype classification earlier which you can go through if you want. I focused on separating gametypes with objective types, as they can be mixed and match amongst different game types. (i.e. a game may feature a CTF component as well as capturing/destroying some objective.)
The objectives I've listed:
Flag/Flag Capture Point: Fairly straightforward but important to note that flags and capture points can be mixed around. (i.e. neutral flag CTF where teams fight over a flag in the middle of the field.)
Container Target: Represented by any container that can hold water (i.e. water bottles, buckets, etc.), any team that fills up an opposing team's Container Target completely using water will score against it.
Attack Target: Similar to the Container Target, but easier to score against. It is any target that can be soaked or blasted by water directly, such as a square of toilet paper. When the target is soaked or destroyed, it is considered scored.
Bomb & Target: Basically the bombing game in Halo, Counterstrike Source, Call of Duty, etc. There are different ways to represent a bomb and to time it though.
Your objective idea is probably a better substitute for the Attack Target.
As for counting hits against players, that is a different matter, as it is not the same kind of objective as what I mentioned above. The only way hit rules and objectives tie into a gametype is when hits count as points which can contribute to a team winning or loosing (i.e. get a kill = 1 point, capture the flag = 5 points, as opposed to standard CTF where kills are not counted, other than having teams share a pool of limited lives), as well as objective completions.
Invulnerability is something I've some experience with, particularly with infection. My approach is a lot different though; the defending team (or survivors in Infection) are vulnerable but get as much waterpower as anyone could ask for (two hoses, CPS's, etc.), and the attacking team is invulnerable but is limited to piston pumpers and less. (Water bottles, squirt guns, etc. are also acceptable.) The idea is the same however; make one team invulnerable and balance the game setup accordingly, then allow the teams to swap roles. However, this works best with evenly numbers of players on each team instead of using staggered/unbalanced team numbers. That said, if you divide the teams accordingly, you can still have teams swap roles and give everyone a chance to play each role. For example, if you have 12 players for an 8v4 assault game, you can have several rounds where groups of 4 take turns playing the small team. Essentially, you are dividing the whole group into 3 teams of 4 each, and have the team of 8 simply be an alliance of two such teams. The group of 12 could obviously also be divided into 4 groups of 3 each, making for 4 rounds of 9v3.