Would Shadow The Hedgehog and Sherlock Holmes get along.
I would say that depends on which version of Shadow and Holmes we're talking about, so I'll be jumping between them lol
I can see canon Shadow (and the other members of Team Dark) being an Irregular of some sort, since he is a G.U.N. agent and part of his job is retrieving info (as seen in some of his missions over the years). He also has done that kind of stuff for private parties (the Chaotix, for example), so I can totally see him do it in behalf of Holmes (acd). Maybe he would also help out planning ways of catching criminals and stuff. I don't think they would be anything more than acquaintances though. Both characters have a tendency to not be actually close to more than one or two people, so yeah.
Now, if we were talking about canon Shadow and BBC Sherlock, that would be a little more complicated lol. If they ever ended up working together, it would be because of some external factor forcing them to do so, not voluntarily. I would say they would be kind of cordial to each other, as they would recognize the other as the best in their field, but they would also regard the other as extremely annoying. There would probably be alot of bickering between them and I would not like to be in a room with those two.