Edited to incorporate Marc's comments below:

1) As Marc points out below, the cocompleteness of LRS is Prop 1.6 in Demazure-Gabriel's *Groupes algebriques*.  

2) 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. 

3) For colimits the answer is yes: The proof of Demazure-Gabriel's Prop 1.6 shows that the inclusion $LRS\subset RS$ preserves limits (even better: creates them).

4) Yes. See (2), in this situation (co)limits are constructed via their projections to the underlying topological space.

5) See (6).

6) Colimits are not preserved by the inclusion $Sch \subset LRS$, see Emerton's comment here: [http://mathoverflow.net/questions/9961/colimits-of-schemes][1]. Colimits are preserved by $LRS \subset RS$ as stated above, and so are finite limits by results in 5.1 of loc. cit. 




  [1]: http://mathoverflow.net/questions/9961/colimits-of-schemes