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When I first heard someone say "shriek" for "exclamation point" I broke out laughing (certainly interrupting an explanation they were giving). It's such an evocative word, which one easily forgets as a mathematician repeating it frequently (e.g., the person explaining it to me had never quite thought about the word choice). I also appreciated that in our science we have a much more poetical term for this punctuation mark than in poetry.
Doing it right is a non-trivial issue, discussed in some detail e.g. in arxiv.org/abs/1105.4857. But note that just defining it naively is easy: Nagata compactification exists for derived schemes as well. The analysis that this (homotopy) doesn't depend on the compactification essentially follows from the analysis in Section 6.2.2 of loc. cit.
I'm afraid there's not really more to say than you've said. You have descriptions for open embeddings and proper morphisms and that's enough to describe it in general (say, for schemes finite type over a field). But $j^*=j^!$ doesn't commute with products (for $j$ an open embedding) and therefore doesn't have a left adjoint. Do you have a more precise question in mind?
Then an alternative: my memory is that e.g. Borel's book uses quasi-projectivity but very sparsely. You could try to remove the hypothesis for yourself by modifying the proofs. I think he probably uses it in ways that modern science would avoid via Thomason-Trobaugh.
The restriction to the quasi-projective case is unnecessary. E.g., modern descent techniques for derived categories allow you to pass from the affine case to the general case. I would be very concerned about all six functors existing for unbounded complexes with holonomic cohomologies, but bounded below seems okay (more generally, homotopy colimits of bounded holonomic complexes should be okay, and that's probably as much as holonomicity buys you for free). RH is fine for arbitrary complex varieties (since it's Zariski/etale local).
