Just to address some of the problems you mentioned:

> Lorries getting stuck in them

This is obviously subject to fixing it properly. Adding some vertices to the lorry so this particular stucking cannot occur.

> Them being the weakest part of a towers. Towers looking utterly stupid and useless without them (doors stop working?).
>  Pretty much not being able to hit buildings with flints anymore once the fundament is gone

This is also a reason why I want (as written above):

> I'd pretty much prefer something that cannot be destroyed. Meaning the basements delivers any damage to the building and the building is always destroyed before anything happens to the basement.

So you cannot destroy a basement without destroying the building. We can easily make non-structural buildings weak enough so this doesn'*t become a problem.

> Building sites getting abused for spamming fundaments

With our changes already present in OC, this is easily avoidable. First of all, if you create a construction site in the current state of the game (meaning default tip), you wouldn't get a basement. Only the wireframe construction site that does not interact with any other object in the game. If you wanted to 'spam' buildings and therefore spam basements, you'd need to provide the needed construction material (all of it). A fair trade in my eyes.
Also with the new flags we can simply forbid constructing in the enemy flag radius.

Zapper took on the two examples I mentioned above: flagpole and elevator. I'd say for at least these two it's vital to have a basement. The elevator is otherwise easily jammed (one earthquake is sufficient). Also, falling flags do bear the risk of bad consequences; energy loss, loss of controlled space. An enemy might seek to just move your flag away if it's not possible to steal/destroy it.
The sawmill is also bad for the reason I said above.
In addition, it's very easy to effectivly jam a building, especially the wind generator, by accident. Just dig at the wrong position and it's over.

> Didn't we declare that we want to actively support tilted buildings? Why do we need straight floors?

It's still my plan to somehow include rotable construction sites into the game. Thus you could simply construct your building tilted.

> Do we have granite chunks/barrels or something similar? And what about bridge segments? When in doubt, I would favour trying to make these more useful. I think there's plenty unexplored gameplay space here.

That's due to conceptual work. Feel free to suggest something like self-administered basements. But, for the reason mentioned, I'd still strongly support basements coming along with the critical buildings like flagpole and elevator.

Last but not least:

> fundaments

I have to admit that I never heard of the use of this word in this context. It were always the terms basement or foundation as far as I've come across this. But this is due to correction by a native speaker or someone who knows. ;)