I use a Janome J3-24, it’s a great sewing machine and has one of the highest Presser foot lifts needed for many layers or thick fabric.
I also have a Singer Heavy Duty 4454 which is faster and great for thinner layers, but prefer the Janome
Fleece - Tia Knight.co.uk by the meter, spend over £25 you get free delivery which is really quick by DPD
I don’t like waterproof liners so use:
Top - anti pill Polar fleece
2nd layer - Polyester sheet wadding 6 oz
3rd layer - plain white anti pill polar fleece
4th layer - anti pill Polar fleece
the polyester sheet wadding works really well to wick away any pee, it dries really fast, keeps its shape at around £6 per meter by 60” wide - ebay, that’s free delivery too
Lots of instructions for tunnels, cuddle cups, cubes on Utube and Pinterest