Is this somewhere I can bring my 2 year old tubbing with me with him on my lap

I think you might want to call to have that question answered. I'm not sure what their policy is with lap riding

Any restrictions on who can ride and how are based on height. Anyone 42" or taller needs to purchase a full priced ticket. Anyone under 42" with their shoes on get in for free and can ride in the lap of a responsible paid adult.

Yes on property right next door

I did that it with my son at 3 years old 2 years ago. Unless it changed you should be able to.