Would upgrading my basement with sandstone blocks be worthwhile?
You have a couple things going on here:
1) The posts.   an engineer would have to evaluate to be sure, but it's most likely that you will not be able to remove the posts without providing beams of some sort to span the space between the walls.     Their irregular arrangement probably lines up with walls that they are supporting in the first story, which in turn are supporting the floor of the second story.
2) Clearance.   Digging deeper to get additional clearance will be challenging, because you will have to leave the footers under the existing foundation in place, and not undermine them.   This means you will likely end up with a floor that is a fair amount smaller than the existing space.    Also, you will have to support the part of the house that are held up by the posts while they are being relocated/lengthened.

In either case, you will need advice from someone knowledgeable in building structure and foundations.

Overall, it may be easier/cheaper to raise the level of the house and add additional height to the existing foundation (if it is in good enough shape to take the additional load) than to dig deeper.   A contractor that is experienced in this will know how to support the house while it is being raised, and you may be able to include new support beams to eliminate some or all of the posts at the same time.