Try going to the place where you bought, got or stolen your current horse. Shadowmere should be nearby. Try casting the Alteration spell, Detect Life, or use the Aura Whisper dragon shout (longer range than Detect Life) to try to find Shadowmere. Shadowmere should be within the range of Detect Life or Aura Whisper.
If not, Shadowmere might have gone back to the place where you first got him: the pool in front of the Dark Brotherhood sanctuary entrance, in Falkreath Hold (near Falkreath, the city).
If you still can't find Shadowmere, try teleporting him to your current location by using console commands (make sure you are outside [a location you can Fast Travel away from] and not in a walled city when doing this, as horses are not meant to go inside walled cities or other indoor locations, and may cause glitches).
Console commands to teleport Shadowmere to the player's current location:
prid 9CCD8
moveto player
'prid' targets an entity - Shadowmere in this case. '9CCD8' is Shadowmere's RefID. 'moveto player' moves that targetted entity near the vicinity of the player.
If Shadowmere is still not near your vicinity after doing this, he may be dead (that or his corpse was teleported near you). Try these commands to bring him back in the game and resurrect him:
enable
resurrect 1
More info on console commands here: http://uesp.net/wiki/Skyrim:Console
No, none of the Dark Brotherhood's quests intersect in any meaningful way with any other quest chain. Some targets may have minor 'radiant' quests available, but none are significant in any way.
The worst thing that can happen is you can end up with a very large bounty on your head, which could make completing other quests... difficult at best. However, this is both a solvable, and preventable problem.
The only collision other than the one you've noted, is that
Vittoria Vici, the target of the main-line quest Bound Until Death is also involved in the incredibly minor quest The Spiced Wine.
Best Answer
Without Cheats
The simple answer to your question, without cheats, is no.
The game is specifically scripted to give you Shadowmere once you have completed the Dark Brotherhoods quest related to the horse.
With Cheats
With cheats, you can get Shadowmere however. First open up your console, and type:
help ShadowmereIt should come up with an "ID" for the NPC for Shadowmere, this is what you want to spawn in.
Then, type in:
Shadowmere will spawn in next to you, now select Shadowmere by clicking on him/her/it while the console menu is open. Now type in:
And Shadowmere should be yours.
Conclusion
I understand that you want to play the game legitimately, and I'm like that too. But if you really want Shadowmere, perhaps a good way to "roleplay" it would be to say to yourself that you found him/her/it while destroying the hideout and tamed him/her/it? Up to you.