Ok. Lots of problems. Don't panic. 
The washing machine could be as loud as they say. I've been in a downstairs flat before with poor build quality and low soundproofing, where it sounds like an earthquake when the neighbors upstairs washing machine was spinning. It vibrated the walls. It's not your problem though it's the neighbors problem. You're entitled to use your washing machine. Do your best to use it at daytime hours, if you can. Ignore comments from neighbors.
Kids running in and out - it isn't your kids. You will have invoice from childcare for hours they're there as evidence it's not you kids, which you can show to officials if necessary. So ignore neighbours comments.
Who's letting agent is contacting you? You mentioned you're HA, so you won't have a letting agent then 
? If neighbors are private rental and have a letting agent you don't need to take calls from this organisation, block their number. Don't feel obligated to listen to moaners.
Washing lines, 6 lines and 6 flats? Contact your landlord and find out officially which one is yours, use that one and no others. If people choose not to use theirs, that's up to them and doesn't mean you can use it. If they was friendly and didn't mind then fair enough, but they aren't, so don't give them ammunition for saying you're unreasonable.
Allocated parking, buy those things that go in the ground and you lock them upright when you're not driving in or out of the space so it's security for your car when it's parked there and security against people using your space when car isn't parked there. Again, if the spaces are not labelled find out from your landlord which space is officially yours. Ask for permission to install the lockable post things if you want, but personally I'd just do it and if landlord asked why I'd explain problems I'd been having. You don't have to suck it up others using your parking space, you're paying for it as part of the rent.
Recycling bins aren't your problem. You have a car. Find other local communal recycling bins in your area (eg supermarket car park?) and take your rubbish to those. If it comes to it, take your refuse to the tip yourself, if that bin is in question too, with people putting recycling in it. No, you shouldn't have to, but it's a solution you can use to prove its not you causing the bin contamination. Then if anyone official complaints to you explain you're not even using the flats recycling (and general refuse, if applicable) bins at all. Ignore neighbors comments.
Hearing the children running round playing inside their homes is life. Yes it's intrusive if you live below it, but it's life and neighbors have to suck it up. Slammed doors etc is lies to make out you're a problem tenant and get you eviction. But they'll need evidence, sound recording etc, and they won't get it because you're not doing the noise! So ignore comments and don't worry.
Headphones on when you go to and from home/car (switched off if you need to hear kids) don't get into conversation with neighbors blank and pretend you can't hear them. Big visible headphones. Smile and wave if you can't blank, eg if they're in front of you, but don't get talking. You're not obligated to listen to moaners or interact with people you don't like.
Invite your landlords to leave camera/sounds recording in your home/near bins/by main door etc if you get your landlords on at you, maybe believing neighbors. Calmly refute neighbors lies. Say how stressful it is for you to be harassed by neighbors this way.
Check who is at door before answering, if it's neighbours then don't answer. You don't have to stand there in doorway letting them hassle you. If they start rudely hammering on your door trying to get you to answer, record it and report it to your landlord (and police if applicable).