Lazard's theorem for connective modules over connective ring spectra is equivalent to Definition (2) (Theorem 7.2.2.15 of Higher Algebra)

See the discussion on page 15 of arxiv.org/pdf/2008.00330.pdf

It will contain at least all these: map.mpim-bonn.mpg.de/Fake_complex_projective_spaces

See Proposition 4.2 of arxiv.org/pdf/1110.5851.pdf or Proposition 1.10 of people.math.rochester.edu/faculty/doug/otherpapers/…

Maybe try "From spectra to stacks" in the TMF book (people.math.rochester.edu/faculty/doug/otherpapers/TMF-DFHH‌​.pdf)

I believe abelian categories are always Ab-enriched, see nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/…

Another example worth spelling out (I guess it is a special case of (2)): the derived category of comodules often has the dualizable comodules failing to be compact. Fixing this gets you (more or less) Hovey's stable comodule category (arxiv.org/abs/math/0301229)

Very much related to the previous two comments - I would try seeing if you could get something out of arxiv.org/pdf/1507.06867.pdf (which doesn't consider BP, but does consider MU).

Take $F_pC_*(X) = C_*(X^p)$

(I got my adjunctions the wrong way around in the previous comment, I meant a case where $f_!$ is not right adjoint to $f_*$)

@Gabriel In any case, if you are not aware of it (although you probably are) the paper by Balmer--Dell'Ambrogio--Sanders should be of interest (arxiv.org/pdf/1501.01999.pdf) - Equation 3.7 is what you want, although they are working under stronger conditions than what you want. But maybe you can find something useful in there anyway.