The simplest natural example is to use the van Kampen theorem (in Ronnie Brown's version) to work out the fundamental group(oid) of the circle. The details are in his book Topology and Groupoids (http://groupoids.org.uk/topgpds.html). That source also discusses colimits of groupoids in this context. If you look around on Ronnie's home pages (http://groupoids.org.uk) you can find discussion of many of the points that you raise.
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