I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb{Z}$-one? how can one denote it?) is zero (in all degrees). My questions are:

1) Is it ok to call these spectra acyclic?

2) Does there exist any "description" of all acyclic spectra?

3) Is it true that the ($H\mathbb{Z}$-)cohomology of any acyclic spectrum vanishes? Possibly, this fact can be deduced from Proposition 16.2 of the book, Margolis H.R., Spectra and the Steenrod Algebra: Modules over the Steenrod Algebra and the Stable Homotopy Category, North-Holland, Amsterdam-New York, 1983; yet I am not sure.

4) Is it possible to localize $SH$ by the full subcategory of acyclic objects (so, do we obtain a category whose morphism classes are sets this way)? If this is possible, then we would obtain a "better $SH$", and this should contradict a result of Schwede (on the Margolis's axiomatisation conjecture); yet I am not sure in this argument (see the Upd. below).

Did anyone consider this localization?

5) Can one describe the left or the right orthogonal to all acyclic spectra, i.e., the objects that are only connected with acyclic spectra by zero morphisms? Note in particular that there are no non-zero morphisms from acyclic spectra to connective ones.

Any hints or references would be very welcome! A related matter: I am interested in texts that treat Atiyah-Hirzebruch spectral sequences for arbitrary spectra.

Upd. So, 3 is fine; thanks! Is the converse implication true (are spectra with vanishing cohomology acyclic)?

About 4: note that $SH$/acyclic spectra contains the category of finite spectra (and the category of connective ones also). So, why does not one consider this localization as a "reasonable" substitute of $SH$?