Does Grp have a (co)limit for every diagram? If not, what properties does a diagram need for there to be a (co)limit? What is an example of a diagram that doesnt have a (co)limit?
The relevant terms here are completeness and cocompleteness.
A quick search for these terms will yield the following information and more:
A category is complete if it contains all limits over small diagrams (where the index category is small) and it is cocomplete if the colimit of all small diagrams exists.
Indeed the category of Groups is a bicomplete category meaning both complete and cocomplete. This follows from the facts that it contains all pull-backs, push-outs, (small) products, and (small) coproducts. The existence theorem of (co)limits takes care of the rest.
Thus the kind of diagrams for which there may not be a limit must be indexed by a proper class of objects.
As a side note the reason one considers only small index categories in the definition of completeness has to do with the fact that asking for completeness with respect to all diagrams would be a very narrow definition. In fact only thin categories can satisfy this property.