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Edited to incorporate Marc's comments below:

1. 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.

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. 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.

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

5. See (6).

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.

Edited to incorporate Marc's comments below:

1. 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.

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. 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.

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

5. See (6).

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.

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: Colimits of schemes. Colimits are preserved by $LRS \subset RS$ as stated above, and so are finite limits by results in 5.1 of loc. cit.

Edited to incorporate Marc's comments below:

1. 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.

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. 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.

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

5. See (6).

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.

Here's some of the answers. I'm skeptical that (1) is true, but other people seem to claim it's the case (with no reference).

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.

3. For colimits, I believe the answer is no with an example as in (6) below.

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

5. Colimits are not preserved by either inclusion. For the first see Emerton's comment here: Colimits of schemes. For the second, consider the pushout of a diagram of points with fields for their structure sheaves. In ringed spaces the pushout is a point with the pullback of the corresponding rings, which won't be local.

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: Colimits of schemes. Colimits are preserved by $LRS \subset RS$ as stated above, and so are finite limits by results in 5.1 of loc. cit.