I fear I lack the self-confidence to go outside at my current 'state'
It'll take time for that to build, I know it's uncomfortable and I had a few days where I felt my posture/breathing go to crap because I felt anxious or like people were watching. I zone out now, nobody is watching or caring - go for it and tune into your little world.
With C25K getting out the front door is half the battle!
Consider pushing that comfort zone hard in the face as the longer that drags on, the more it'll start nagging you and suck esteem.
Do something that scares you. Even a walk around the block as it's a success!
and I really don't know HOW to carry all this stuff
I found messing around with stuff created procrastination and excuses. The water isn't essential for short runs/walks - just make sure you drink enough during the day. Keys? Consider a hidey hole near the flat if possible like a plantpot (not for everyone but one less thing to worry about) harness or strap to hold the phone, or invest in a smartwatch/tracker.
If you decide you'd like to carry stuff, strap on a running bag or belt - you could even get a hydration pack with hose. There are options :) (karrimor so some decent belts/bottle belts/bum bags)
Posture
Don't get sucked in to overthinking and complicating to the point you get annoyed - you're as unique as everyone else so there's a balance of good posture and natural gait.
Think tall, hips+chest forward and shoulder blades back. Find a posture that lets you breathe easy while not "forcing" muscles or you'll get sore and think "I'm doing it wrong"
This also takes confidence to carry as it feels wrong and takes time to develop - keep it simple and it will naturally evolve as you gather core and leg strength. Walking is good practice and really good for strength.
Stretch afterwards and build your own set - I started with quads, upper and lower calves before tacking things on like hip stretches and planks etc.
Over time you'll get a feel for posture and what your body is actually doing - when you get confident in what you're doing there's no harm in doing gait analysis or getting someone to watch you/record so you can see yourself run. There's also physio and coaches/trainers
Practice wall posture and exaggerated hips forward and back, go way too far back on shoulders and too far forward - find a comfortable balance.
I think maybe I'm not doing it right
I only think that every day.
Salt
I probably overdo it but I started to get braver with mixes of foods, I like salted roast nut mixes and keep snacks going in moderation for simple carbs like mini pretzels or popcorn.