What are the Splatoon game modes and how many players does each mode support?

The only thing you can do with two players, is an offline balloonfight in the Battle Dojo. Not the single player story mode. No turf wars.

You cannot play online at all with 2 local players. Not in co-op, turf wars or ranked.

What are the local multiplayer game modes for Splatoon and how many players does each game mode support?

For local multiplayer there is only one game mode and only allows 2 local players. Player 1 uses the gamepad's screen and Player 2 uses the TV screen to compete who can pop the most balloons. You can totally use this to compete for kills instead or other rules you make up with your buddy instead.

What are the online multiplayer game modes for Splatoon and how many players does each game mode support?

There are several online game modes in Splatoon, and some are unreleased. All modes are 4v4 with automatic team matching. Manual teams coming in a future update.

• Turf War - Compete with the opposing team to ink the most turf. Your team wins based on % turf covered vs opposing team's % turf covered. This mode is unranked meaning losing has no consequences, however automatic matchmaking will place you in lobbies with similar play styles and skill levels.
• Splat Zones - Kind of like King Of the Hill. Each map has 1 or 2 zones that must be ~80% your team's color to be considered in your control. If you hold control of the zones for 100 intervals of time (~50 secs) you win via Knockout. Otherwise the in game timer will determine the winner after 5 minutes based on which team held the zones the longest. This mode is extremely intense compared to the lax turf wars. Winning/losing affects your rank.
• Tower Defense - (This mode will be released in a future update in August) If you've played TF2 this mode is similar to the Payload mode. There is 1 tower that moves on a rail towards the spawns. By standing on the tower, it will move in either direction depending on what team you are on. If it reaches the end, your team wins, otherwise when timer is up, which-ever side the tower is closer to wins.
• Rain Maker - (Another mode to be released in a future update) Similar to CTF in other games but you will be competing with the other team to bring the "Rainmaker" to your base. It can shoot Inkzooka strikes but makes you move slower. Not much else is known about this mode at the moment.
• More? - Nintendo has said that they will constantly add content to this game for a while. It's possible more modes are on the way.

"Does it offer online coop?"

Not sure what this question is asking. For the single player campaign? No. For online as in making your own teams with friends and having "clan/guild wars": Yes, but in a future update.

"If so, how many players?"

4 Friends can form a team

"Does it offer offine coop?"

No, only competitive play with 1 other player.

"Does it offer other online modes of play?"

Offline modes of play include the single player campaign, the amiibo challenges, the 1 on 1 local competitive play, and the arcade games (minigames)

Nintendo said that they will be actively supporting the game and constantly provide updates so they will most definitely add more modes and features in the coming months. Already, we are getting 2+ new weapons per week, and new stages every other week. In August they will provide a huge content update which is confirmed to have new modes, creating your own teams, and new weapons/customizations.

Closing words/Personal thoughts: I am simply in love with this game and recommend it to anyone who has a Wii U. I find myself wasting days on end playing it to figure out new strategies and ways to play. It's a really refreshing game and I'm excited to see more!

(PS: Thanks for your help on SO!)