100% send the pets to cattery/kennels/pet sitter for the day if you can afford it (stressful having them underfoot for the removers, you'll then have more space in the car for stuff, and if your cat(s) are anything like mine they will take any opportunity to 'escape' for the day to avoid being put into their carriers which is the last thing you'll need 
).
For the kids, I would try and get them excited about the new house, can you let them choose paint colours for their new rooms and pick out new bedding/furnishing etc., or things to play with in the new garden? Kids are very resilient so I wouldn't worry too much about them being upset about losing their house, so long as they don't see you being too devastated they will probably treat it all as a big adventure. I wouldn't try and get them too involved in 'helping' on the day as at their ages that will be more trouble than its worth (although they can help to pack up and sort through their toys beforehand), if you can't send them off to a friend/relative/childcare then I'd set them up in a quiet room with ipads/snacks/telly so they are out of the way while the packing/removals are happening and then same again on the other side.
If you can call in any favours from friends/relatives to help out on the day, especially people who are strong and/or have a car that can take any 'overflow' from the movers, that would be ideal - don't be shy about asking for help, this is the kind of thing people like to help out with IMO.
I do think it's best to use the opportunity of the move to get rid of as much stuff as possible, the financial and 'effort' cost of packing up/removing/storing/housing things you may or may not need will probably outweigh the risk you may throw away something you later need to replace. Obviously this doesn't hold for things with sentimental value but you could use some Kondo-esque methods of taking pictures of things so you have the memories rather than needing to keep the physical object, or picking just one or 2 items out of a collection, rather than everything?
One of the best moving tips is to make a survival pack for your first night(s), in this I would put mugs, tea/coffee, cereal and some UHT milk, enough plates/bowls/cutlery for your first nights dinner and first mornings breakfast (disposables are easier/take up less space), phone chargers, bedding, essential toiletries, essential medication, a small quantity of pet food and any precious toys or comfort items the children need to sleep. Then do not let this bag/box out of your sight, don't put it on the van but take it in the car with you. Then you know that whatever delays happen to the van, even if you aren't able to unpack a single box on arrival, you have every thing you and the DC need for that first night and the morning after. I'd budget for a takeaway or chippie tea that first night if you can so you don't have to pull out your cooking stuff, if not then maybe cook and freeze something like a lasagne or shepherds pie which can be defrosting for the first day, then popped into the oven when you get to the new house?
Good luck, you can do this!