soft question - How much of scheme theory can you visualize?

Well, you asked 10 different questions, and I am not sure what you mean by "nonproper" ($Spec A$ is not proper). But let's see.

A scheme is a very geometric object, with practice - or maybe just habit - one learns to visualize it quite well. If you already see geometrically $Spec$ of a finitely generated algebra over a field $k$ (including algebras with nilpotents which you visualize as "thickenings", including $k$ not algebraically closed which you visualize as Galois orbits; you looked at these, right? these are important steps) then you are almost there. Add some other standard examples such as $Spec(mathbb Z)$, DVR, a double-headed snake (the first nonseparated scheme), and you already know plenty do start doing research.

Infinite-dimensional algebras? Well, I suppose it is just as hard or easy to imagine them as infinite-dimensional spaces.

The fiber product is a perfectly geometric notion as well, and fairly easy to visualize. You begin by looking at fiber products of sets and you progress from there through some standard examples. Isolating a fiber of a morphism is an important case. And then look at some examples where the residue fields of the scheme points change. Learn the simple way to compute the tensor product $Aotimes_R B$ by using generators and relations of $A$, and you will be up and running in no time.

As far as the balance of geometry vs algebra, I suppose that depends on a person and everybody is different. My advisor used to say that geometry comes first and then later algebra follows, and I tend to agree. I think you get nowhere without geometric intuition.

But if you are serious, at some point you will need a solid commutative algebra foundation. Fortunately these days there are plenty of nice books, starting with the very nice and elementary "Undergraduate commutative algebra" by Miles Reid.