Your reactions to posters does seem extreme - would it be fair to say you can have a quite 'explosive' or 'prickly' attitude?
Having a bunch of clean clothes pulled on to the floor makes more work for you - it might be easier in the long run to put them into a basket and take them upstairs whilst your DC are safely behind the baby gates you mention, as then you aren't doubling your work by having to pick it up multiple times.
Extra bin bags - that's something that would mean the foxes pull everything all over the street here, especially nappies, which is disgusting for everybody. In other areas, it would mean the rubbish wasn't collected at all.
If there are lots of random things out the back, it does really help to get them out and cleared. Doing it bit by bit does work in the long run, far better than having to schedule a one off bag and tip run, although if there's lots and lots of things, it might be better to do it all in one go - but it sounds as though you don't have the time for that. Doing it bit by bit is therefore easier than not doing it at all.
Shoe racks - well, if they aren't by the front door, they might as well not be there, in my opinion - you may as well take all the shoes upstairs and shove them in the bottom of the wardrobes so nobody has to look at or risk tripping over them. And again, it was making more work for you than dealing with it.
But children do need their hair brushed and whilst a bit of dirt doesn't hurt, having a place that looks dirty and messy isn't the same as that.
Your angry reaction reminds me of my mother's and a few old neighbours' excuses for having a dirty and dangerous home and garden. It was always 'I'm too busy', 'It's the kids' fault', 'I don't have time or the money like x person', 'I'm not rich enough for a cleaner', 'It would be tidy if it was bigger', 'It'll be fine once...' (which would be a situation where it was even less likely to be fine, such as when there are two children to wrangle into different places, getting to the childminder and work and still doing all the home stuff as well because the other adults in the house isn't around to do it).
It can be very helpful to have clear surfaces and spaces, as there aren't 'things' demanding you see them and deal with them. It's a lot calmer to sit down where you aren't thinking 'there's the washing, there's the shoes, there's the dirt on the kitchen cabinets, there's the washing up', etc, I'd also suggest that if at all possible, you get a dishwasher, as being able to shove everything in there and go and do something more interesting makes a huge difference to both the look of the kitchen and saves you anything up to an hour a day, depending upon how much you use as a family of four.
Getting to the point of demanding people are fired and getting so angry on a mere internet thread does make it sound as though you are using attack as the best form of defence. And maybe you would benefit from having some more help (as you aren't getting much from a DP that's out at work all the time).
Maybe seeing the GP isn't such a bad idea, as they can make sure that you haven't got anything underlying that's adding to your reactions, such as hyperthyroidism? Once they're eliminated, I'd suggest you find ways to get stress out - if a big clearout doesn't work for you because your dad won't come and help drive stuff to the tip whilst DP is at work, perhaps finding an exercise class or something you enjoy whilst they babysit would help?