Questions.

“If you don’t see an invitation on your calendar, does that mean that you aren’t considered good enough to raid?”

No.
In order to have a viable progression raid, raid leaders need to recruit two regular tanks and two-three regular healers. And the right mix of ranged and melee DPS. They try to invite people that have been already going to the raid in order to maintain group cohesion.
We are too big now to post raids that invite every level 80. Also inviting every level 80 to every raid doesn’t work either.
If you would like to be in regular raid and don’t see it on your calendar, then talk to the raid coordinators (RCs).
Snuffy is the the RC for the ICC25. Glori and Avante are the RCs for the two ICC10 progression raids. Mainartin is the RC for the intermediate raid (ToC/VoA/Ulduar). And Prettydead is the RC for the JV Naxx raids. If you still don’t get the invite, then talk to Bob (Atleast, Bigcoww). Also check with your class lead. If you don’t know your class lead, talk to Kaels. If all that fails, then talk to Mama.
Remember that raid balance is very important. And that each class brings something to a raid. A raid with four DKs is not going to be as successful as a raid with two DKs, a rogue, and kitty druid. I will post a guild raiding class breakdown so you can check which classes we need and which we have a lot of. But I can tell you right now that we have lots of DKs, followed by hunters, while we have very few warriors, rogues, and mages. Mixed classes that can like priests, druids and paladins are always good. Just keep in mind a priest that can heal and DPS is more likely to be included in raids than a priest that is all damage all the time.
Most raiding guild require people to play certain specs, but we don’t. However, I can’t change the mechanics of the game.

“If I work late and can’t make the regular 9:00 p.m. (ingame) start time, can I still do progression raids?”

Yes.
But you will have to take some initiative. Glori’s raid should have a later start time since this was started as a west coaster raid. For the ICC25 man, I would encourage you to team up with someone who has a gear score in the 4K range to take the first shift for you, and then talk to Snuffy. Get to know your guildies and work as a team.

A final note.
Anyone can organize and run a raid. If you would like to see more fun raids, perhaps on a weekend afternoon, then post it on the calendar. You will probably have to recruit people but that’s part of getting to know your guildies. If you don’t have quite enough people, you can always include a few pugs.
The morning coffee raid is on hold for now. Mama would like to bring it back soon.
Don’t be afraid to pug raids. There are many small guilds running ICC10 and ICC25 that always need people. This is the best way to gear up alts since we are trying to discourage progression raiding on multiple toons within the guild.

If anyone has additional questions or comments, please post them in the comments.