Edited to incorporate Marc's comments below:
The cocompleteness of LRS is Prop 1.6 in Demazure-Gabriel's Groupes algebriques. Completeness is proved in http://arxiv.org/abs/1103.2139, Corollary 5.
Yes. The category of ringed spaces is (co)fibered over the category of topological spaces (which is (co)complete) and has (co)complete fibers. EDIT: To answer the OPs question, I can't think of a reference offhand but it's not too bad to prove straight from the definitions. Take your diagram upstairs, form the (co)limit downstairs, choose (co)cartesian lifts of the projection/inclusion maps from/to your (co)limit, form the (co)limit in the fiber, and you're good.
The forgetful functor RS → Top preserves limits and colimits. See (2), in this situation (co)limits are constructed via their projections to the underlying topological space.
The proof of Demazure-Gabriel's Prop 1.6 shows that the inclusion $LRS\subset RS$ preserves colimits (even better: creates them).
See (6).
Colimits are not preserved by the inclusion $Sch \subset LRS$, see Emerton's comment here: Colimits of schemes. Colimits are preserved by $LRS \subset RS$ as stated above in (3). The inclusion $Sch \subset LRS$ preserves finite limits by results in section 5.1 of Demazure-Gabriel.
