I will answer your questions labelled as you have them

(a) there is not statistical reason for excluding cohort studies from your analysis but there might be substantive reasons if they are performed on people with a very different case-mix from these in trials (randomised or not)

(b) I think this has a similar answer to (a). Part of the rationale for network meta-analysis was to develop techniques for indirect comparisons between treatments and single arm studies are an extreme case here. You might want to consider whether you can introduce a study-level covariate to account for case-mix if you can see how to do that.

(c) I do not use Stata so cannot comment here.

I will answer your questions labelled as you have them

(a) there is no statistical reason for excluding cohort studies from your analysis but there might be substantive reasons if they are performed on people with a very different case-mix from these in trials (randomised or not). So it is perfectly in order to include a single arm study using a medical treatment and a single study using a surgical treatment. Each arm gives rise to an estimate of effect like the cure rate. Each arm in a trial also gives rise to similar estimates.

(b) I think this has a similar answer to (a). Part of the rationale for network meta-analysis was to develop techniques for indirect comparisons between treatments and single arm studies are an extreme case here. You might want to consider whether you can introduce a study-level covariate to account for case-mix if you can see how to do that. People will always be more sceptical about conclusions if the majority of your information comes from indirect comparisons.

(c) I do not use Stata so cannot comment here.