Is there an ideal evolutionary body?
All mammals have 4 limbs, and all fish have gills and fins (except for some, but I will get to that). What if that is because that is the perfect form for an organism in that kind of environment?


Different mammals species live in very different environments. We all have four limbs (except exceptions), not because it is adaptive to all of us for a specific environment but because there a strong phylogenetic signal. You should have a look at Why don't mammals have more than 4 limbs?.

All fishes have gills and fins (except exceptions). This is definitely because they inhabit a common environment. Gills allows fishes to get oxygen from water. Fins is a general structure that allows swimming. You will note that marine mammals also have fins (we often talk about convergent evolution) Btw, the post Do fish break a water molecule to absorb oxygen? might be of interest to you


  how much does the mindset and disposition of an organism affect its end body shape, and how much more random does it make the process of evolution?


I don't really understand what you mean by "mindset" and "disposition".

Nothing is random except in the consideration of unknowns. One can make sense of a probability only by consideration of a priories. While tossing a coin is often considered as the random event by excellence, a toss of a coin is everything but random if you were to know the exact forces that apply to the coin. So asking is something is random without stating what are the a prioris you consider knowing and not knowing makes no sense (although it may make some sense in quantum physics).


  say biologists were able to create organisms and watch them evolve quickly. If you took two single celled organisms that had no similarities, and put them in the exact same environment, would they evolve to look similar?


We are actually able to witness evolution in a matter of weeks in lab with organisms with short generation time such as yeast or E. coli.

The repeatability of evolution is a vast subject. The issue with such question is that it depends a lot. It depends typically on what evolution you are expecting to see and the size of the population. For example a simple change in allele frequency at a selected locus in a large population is typically very much repeatable. But if you think about larger time scale then, most think it is wise to consider that evolution would take a totally different path. There is a famous quote that says:

"If the tape of evolution were replayed a million times, a species like ours would not necessarily evolve" Stephen Jay Gould

You should have a look at Is evolution a predictive theory?.