Hi Colin
I've just returned from Mulu soI think I can help with your queries.
I would definitely just book it directly through the park, so you can be flexible and custom-make your trip to fit to exactly what you want to. You can email them through their site, which also has loads of good info on it: https://mulupark.com.
You can't go into any caves without a guide, so everything needs to be booked. Do it in advance to ensure spaces don't fill up. If you want to do adventure caving you need a min of 3 people (plus the guide) so if there is an accident 1 person can stay with the injured person and 2 people can go for help). We were just 2 so we booked a way in advance to do Clearwater and they will then try to add additional people to ensure it can go ahead.
You won't be able to do Garden of Eden unfortunately. That is closed permanently as 2 people died there last month when a sudden flash flood swept two people into the cave system. Understandably since that happened they are being very cautious, so if it has been raining a lot it is likely that your caving will be cancelled or postponed. Th guides we went with were very experienced, and caving is an awesome (if intense) experience. But if you don't fancy that just go to the showcases, which are awesome in themselves.
They don't provide lunch but you can buy a packed lunch and snacks to take from the little cafe there.
It is all very easy to navigate - the reception and cafe area is very small and a short walk from the airport. there are a bunch of nice local homestays on the road leading up to the park if you don't want to stay in the park.
You book everything in advance but just pay on arrival. You can use a card or give them cash. NB there is no ATM in Mulu (you're in the middle of the jungle!), so take plenty of cash out before you fly there.
We did 4 nights there, which was a good amount of time although we could probably have done it in 3. That's long enough to see the 4 main show caves, plus do an adventure caving day, and you can do the canopy sky walk one morning before you do the showcases if you fancy it.
The place is beautiful and awesome, a real gem. You will love it!
Stuart :)