Question
I have two questions:  What is your recommendation to prevent squirrels from
eating wood trim on a house and how do you get rid of moles without resorting
to traps or pesticides?

Answer
For the gnawing, you have only three options.
1. Prevent their access to the structure (i.e. trim back tree branches, install porcupine wire)
2. Apply a sticky repellent on the sites you want to protect or some sort of physical barrier like aluminum flashing.
3. Population control.

Regarding your question on moles, you have essentially prevented any practical solution because you have eliminated the common methods. Don't get offended but you do have other options, but you must decide if they are practical. I am just telling you now that it is highly unlikely that they will be. But here we go.

1. Flooding can kill moles.
2. Paving your grass over with asphalt.
3. Install exclusion. Which involves digging a trench around your property down at least 4 feet or below the frost line whichever is deeper. Install 1/4 inch hardware cloth and extend it up above the soil at least 6 inches (sometimes moles move on top of the ground).

You will see a lot of ads touting this or that non-chemical repellent, usually some sort of thumper. Buy one only if you like lawn ornaments. We are not aware of any solid research that supports their use. We know of wildlife control operators who trap moles in yards where these things exist.

Additional information on both problems can be found at http://icwdm.org