1) Set your alarm clock for tomorrow morning
2) Prepare the check, envelope, etc.
3) Call your friends, see if there's still time to join them, make sure you come back in time for a good sleep
4) Mail the envelope whether you're joining them or not
5) Wake up early tomorrow, go to the bank, call your bf

You know what? It's the end of the holidays. It's perfectly acceptable -- even healthy -- to cocoon for a day or two in order to prepare yourself for getting back into the swing of things on Monday. Accepting that, and realizing that you don't have to apologize for it, is how you stop, generally, being angry at yourself.

As for specifics:

It's too late to mail the rent and it's going to be late and my mean landlords are going to charge me a fine.

If you were to mail your rent check today, it presumably wouldn't get anywhere until Monday or Tuesday. Therefore, this problem is solvable. Go to the post office first thing tomorrow morning and send your check out as Express mail. It'll cost you about $16.00, but it'll save you your landlord's fee and ill-will. You could even call/email landlord to give them a heads-up, let them know it's coming via Express mail (because "something came up last-minute that prevented you from mailing it earlier in the week"), which will make you look like you're being extra responsible.

I missed a call and a text from my new bf who wanted me to meet one of his close friends (who's been a little jealous of the time I'm spending with the bf) for dinner. So now they both probably think I am a flake and I missed a good chance to show that I am ok.

You can still show how okay you are to the bf, and to the friend, if the friend is still around. Call bf back, apologize for missing the call, and see if there's still a chance to get together. You don't need to explain, but if he asks, say "Oh, I just had some holiday blues and kind of lounged around this morning. I'm really sorry -- I wasn't avoiding you, I guess I just was kind of spacy and didn't realize you'd called. I'm feeling much better now, and I'd love to meet your friend."

On top of that, I know these are all such stupid problems and I feel so spoiled for being this upset when people have real actual problems with health and money and relationships.

Eh. Yes, there are Big Yucky Problems. But there are also daily trials, and you're having some of those. Don't conflate the two, and don't automatically feel like yours aren't notable simply because they don't fit into the former category.

You know what? All that stuff you meant to do today will get done. And nothing horrible is going to happen if it doesn't get done RIGHT NOW.

Do mail your rent, though. That one's totally fixable.

Thank you.
I cried it out, read your replies, and progress has been made.

Update:
I washed my face,
did the dishes,
straightened up the apartment,
put on deodorant and hairspray so I won't seem so dirty,
got on some real clothes,
wrote an apologetic email to my landlord,
rescheduled dinner with bf and his friend for next week,
put the check in an envelope,
made plans to meet up with out of town guests,
put scarf out of reach of bunny since now half the fringe is missing.

Yeah, note to future rmless: When 5 foreigners want to visit and stay in your apt for over a week, don't say yes, even if you want to be really really nice. Especially when it is around the holidays.

I sent them out to museums today to have a day by myself to get things done and relax, and I guess I just crashed and burned instead. Also not helpful: getting into an email tiff with a good friend and waiting for her responses all day. (That is worked out now though, which is good.)

Now I am going to eat something, then go out, put letters in a mailbox, buy toilet paper (another thing I forgot was on that list), and take the trash out.

Go you!!! It's going to feel so good to get the trash out, and the toilet paper bought and the letters in the mail, that going out will be easier.

Yay! I'm glad it worked out. This was the really concerning thing in your post for me:

I feel so unaccomplished that I don't think I deserve to go out.

Nix! Your face is washed, you have on deodorant and fresh clothes, and your rent is as good as paid - go get em tiger! These are accomplishments in themselves when you're feeling this way, so it's time for celebrations.

Aww, it sounds like you were overwhelmed. You've recovered very well. I hope you're having a good evening!

My family was in town for five days over the holidays. The day after they left? I slept late, didn't shower until 1:30pm, watched hours of really bad reruns, skipped dinner, and went to bed early. I would have done the same thing the next day, except that I had to go to work for part of it.

And my family wasn't even staying with me.

Guests are a lot of work. After guests leave (and sometimes before!) I always need a day or two to just sit, do nothing, and revel in the feeling that *no one* is about to ask me to do anything. Even a fun anything. No anythings allowed.

Paralysis is sometimes necessary. Don't beat yourself up for it.

"I did absolutely nothing today and have no motivation to do any... thing."

Welcome to the past 2 and a half years of my life. Don't let it get to far.

And DO beat yourself up for it if you need to.

Next day update: everything is better.
I mailed the rent and some other letters I've been meaning to write, bought the toilet paper, cleaned the apt, put away the laundry, took out the trash, all of it.

Phew.

I guess I just needed some down time and some emotional release.
Visitors are leaving on Tuesday and then my new roommate is moving in on Thursday and I am hoping everything will be calm calm calm after next week.