I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please tell me a good refernce on theses topics, if there is some.

edit: I was assuming you wanted an equalizer sheaf property, but this is not the definition of cosheaf, see comments  the following has nothing to do with cosheaves then! If by "cosheaf" you mean a covariant functor from the opens of a space to sets/groups/etc., you could look at Moerdijk/MacLane's "Sheaves in Geometry and Logic"  there you can learn some general sheaf theory on sites, which includes the cosheaf case. In particular pushforward and pullback are transport along the the two functors comprising a "geometric morphism". The notion of quasicoherent (co)sheaf can maybe also be defined in this generality by saying that something should look locally like a pullback from the "base topos". Sorry for the jargon, I didn't get from your question what exactly you are up to  just take a look at the book and see if it suits you. 


I've written a preprint about (what I call) contraherent cosheaves of modules over the structure sheaf of rings of a scheme  http://arxiv.org/abs/1209.2995 . These are a kind of dual creatures to quasicoherent sheaves. The preprint will be updated and expanded (eventually). 


If you have the stomach for hard topos theory a good reference is Singular coverings of toposes  M. Bunge, J. Funk The first chapter is probably enough to answer your question. The upshot is that if you have a site, then the category of cosheaves can be identified with the category of cocontinuous functors on the category of sheaves. This should give you a pretty good idea of what operations you can perform on cosheaves. Btw, in this context, cosheaves are also called Lawvere distributions  distributions because of the analogy with the Riesz representation theorem that identifies measures (cosheaves) with linear functionals (cocontinuous functors). Hope it helps, regards, G. Rodrigues 


Look at Bredon's 'Sheaf Theory', Chapter Six: "Cosheaves and Cech Homology" I am not aware of any quasicoherent story. 


According to Skliarienko, the BorelMoore homology (with coefficients in a sheaf) is badly "defective"; and the "right" homology should have coefficients in a co(pre)sheaf. (He makes this point in a few of his works; but most clearly in the editor's comments to the Russian translation of Bredon's "Sheaf theory".) In fact, there exist two papers by Beniaminov which are precisely about this. I'm a bit puzzled that Skliarienko cited them with enthusiasm in early 1970s, but later ignored on several occasions when speaking of the desired cosheaf homology. There is also a followup paper by Golovin which I haven't checked yet. 

