Now I have this image of a blue, fluffy t-rex ripping apart some hapless prey, singing “C is for Cookie, that’s good enough for me!”
Ha Ha Ha… Hilarious!
But seriously, what I think they do is try and associate the features the prehistoric creature with the closest living being and really just warp the features (within the limits of logical constrains of course!) of the present being and make it fit for the prehistoric creature.
Yes. Think of it as the condition at the “base” of the evolutionary tree (or bush). As time passes, and populations evolve, some may maintain this basal condition and some my lose or modify it.
Human embryo’s have gills, tails & what not at different stages that are gone at birth.
Would be a clue IMO.
Human embryos do not have gills. Some embryos develop with laryngeal cartilage outside the neck.
Humans are extinct?
In some sense, we don’t exist. We’ve never identified a type specimen of homo sapiens. 
Our ancestors are.
Developmental stages can sometimes give some clues to the appearance of ancestral forms, but they are not really a reliable guide.
Yes we have. While Linnaeus did not himself designate a type specimen for Homo sapiens (designated type specimens became the standard practice later), he himself was designated as the lectotype (retroactively selected type specimen) by William Stern in 1959.
Generally, “basal” refers to a group of organisms, not a trait. Traits are usually considered “primitive” (or “plesiomorphic”) or “derived” (“apomorphic”).
A basal group is the earliest diverging group within a clade. Many of the traits possessed by that group will thus be plesiomorphies relative to those of more derived representatives of that clade (while also potentially being apomorphies relative to their own ancestors’ characters).
Does it count if a psychologist does it? 
There are three useful guidelines for determining what an extinct species looked like:
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For Quaternary species, what human ‘reporting’ of their appearance was. This can include cave paintings, statuettes (e.g. Sivatherium, the giraffids’ attempt to evolve a moose-like species, was depicted by Sumerian statuettes), legends (the elephant bird Dinornis is known to have been white-freathered from Arab explorers’ accounts), or even contempoarary civilized people’s accounts. (e.g., dodo, Steller’s sea cow).
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What relatively closely surviving species look like.
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What characteristics are common for the econiche and habitat of tje species, even in non-related forms. E.g., forest browser herbivores tend to develop mottled patterning; tall-grass prairie/savannah grazers often have sharply contrasting stripes.
And of course what survives as ‘hard parts’ gives the basic body structure.
It is worth noting that for this purpose, a complete skeleton may be a composite of several incomplete skeletons. Partial skeletons can have enough overlap that we can be sure they are the same critter. We might have one specimen that is the rear part, and another that is only the front part, and yet a third that is the middle, but shares bones with both of the other two, so that between the three we can get the entire picture. As actual complete skeletons are rather rare, in many cases such composite “completes” are all we will ever have.
I once had an amusing discussion with a young earth type who was incensed that the local museum showed a picture of a complete creature when they only had a single leg bone. “How can they possibly know just from one bone?” Well, there is a complete skeleton with a matching leg bone in another museum, so yes, we really know it was this big, and looked pretty much like the picture there. Guy had a PhD in a technical field, but a huge blind spot when it came to anything that might call biblical accounts into question.