I'd say London Plane except for the seed pods. Otherwise, you're requirements won't get you what you want, and you can wait for Zelkova or gingko for the next generation. Maybe Freeman maple altho overplanted now.

Dan

I agree with that suggestion for london plane or sycamore or california sycamore because I think deep down you want some fast growing tree to replace the elms but lets take a closer look at what your requirements are.

*Something that will (relatively) quickly get to a similar, large size.

Sycamores are amongst the fastest growing trees you can get. It is likely the most desirable of the rapid-growers that you can support in your climate. It combines the long life, ornamental appeal (bark) and stronger wood than other rapid growers.

* Something that can (after a few years) support one side of a hammock (our hammock currently hangs between the two trees).

You're not going to find a lot of support to hang hammocks from trees here but the bottom line is say you're able to find a 2" caliper tree and in 3 years, it's 4". That's still not really strong enough to support a hammock.

Lets say you start out with a 4" tree. Well in 3 years, that tree will finaly be getting re0established. Look into a hammock stand. Then you can have your cake and eat it too.

* Doesn't drop sap, slime flux (like our elms), seed pods, or other things that are unpleasant to have on the ground.

Sycamores and planetrees ARE MESSY TREES.
They are very beautiful and IMO very desirable in the right spot but they leaves of the sycamore are the size of a dinnerplate and they do drop some throughtout the season. You also want to make sure you are not allergic to th fine hairs that are present on the leaves. Once they get to about 4" in size, the bark starts peeling off which is the tree's signature feature and very beautiful. As the trees get larger, this will mean more and more bark stripping throught the season. Again, you will have to weigh the benefits against the clean-up. I myself have opted for the cleanup. I have 2 sycamores and 1 london plane at my home.
* Ability to take a good amount of moisture: This will live in our lawn.

Sycamores tolerate moist soil and prefer it. However it is best, not only to encourage drought tolerant trees, but to not waste water, that you adjust your irrigation so that you water your lawn to no more than 1" per week, only when it needs it. I can't imagine anything needing irrigation in your area but maybe it does.

* Preferably something that loses its leaves in the Fall (in California), so that sunlight will come through unabated in the Winter.

Sycamore will do this. OOoohhh, it will do this!

* Good amount of shade in the Summer (for the hammock).

Sycamore is more of a filtered shade. While the leaves are huge, they are not the most densely foliated trees. So this may not meet your req. On the other hand, it means you can grow a lawn underneath it.

* Preferably something climbable for kids (who are now ~2 years old).

There are not too many temperate trees I can think of that are truly good climbing trees. When I think of good climbing trees, I think of southern live oak and banyan trees and ficus like we had in florida when I was a kid. In maturity, sycamore will have a magnificent structure there but you may need a very large ladder or bucket truck to get your kids to those branches.
I really am not sure about what good climbing trees your climate can support but you may look at some of the tropicals for this and those tend to be evergreen.
Gingko Biloba CAN BE excellent climbing trees in maturity but those are slow growers and very slow to establish. Maybe in time for your kids 102nd birthday

Let us remember that fast-growing trees are generally not recommended in urban areas. Why? They are messy and break easily.

It is a very common question by novice homeowners to request a "fast-growing" tree and have all the requirements as we see above.

All the requirements seen in the typical request above cannot be met. All residential trees are a trade-off. You must make your choice.

Lastly, London plane and CA sycamore are allergenic trees as well as messy and tough as nails and adaptable. I loved my large plane in Sacramento, but raking up the leaves abscised by anthracnose every summer killed my allergies.

Dan

California White Oak a/k/a Valley Oak (Quercus lobata).

Metasequoia glyptostroboides. Grows fast easy to climb extremely strong minimal leaves to cleanup due to their size. Check it out and let us know.

what about Chinese Pistache (male only)?

Catalpa is fast growing, fairly strong, and excellent for climbing. It has seed pods and flowers that drop as litter however. I prefer the southern variety, bigxxxx over the northern variety specxxxx, but the northern is a bigger tree. They both have a similar look. I would rate these trees as # 1 for a combo of adaptability, fast grow, and tree climbing.
Sycamore is probably faster growing, but a messier tree in my opinion.

The oak species suggested is a nice tree.