Roaming has become a popular addition to the metagame strategy, wherein one player does not lane nor jungle, but rather simply moves through the jungle/river to initiate ganks on lanes from level 1 and onward, until the teamfight portion of the game begins.
A roamer essentially guarantees that you're either running a solo lane, or two solo lanes if you have a roamer and a jungler. I've also seen a double roamer game played, though this tends to be less effective as enemies who are wise start playing defensively and reduce the effectiveness of your roamers, which put them behind in both CS and levels.
The qualities of a roamer can vary, but the general requirements are that they are mobile, can do burst damage, and have a strong stun or position altering (push, throw, etc.) ability. Taric, Evelynn and Alistar are all very popular choices for roaming. I've also seen Sion and Blitz play roaming effectively, but they were both out of a lack of a jungle spot on the team and normally suit better in a lane.
This strategy is similar to the "double jungle" mindset in that you get three solo lanes, with the exception that there is extra pressure on getting early kills to succeed. A definite risk vs. reward tossup, as successful roamers will quickly catapult your laners ahead in level / CS as well as demoralize your opponents, but unsuccessful roamers will fall just as far behind due to the lack of their presence in the laning phase.
Stealth can only be broken by a stealthed champion performing an action that breaks stealth.
Though not viable by any means, before the sunfire cape's damage was a unique passive, it was fully possible to simply buy multiple capes and follow an enemy champion around while stealthed, relying on the damage to slowly kill them.
In this case, Time Bomb reacts the same as if you cast it on a ward - because the target is stealthed, the effect won't be visible, nor will it break stealth upon detonation. The enemy would just get a sudden burst of damage with no apparent cause.
Best Answer
So long as the ability was traveling towards him when he completed his ultimate and teleported across the map, it will chase him down. The same can be said for Summoner Spell Teleport, Twisted Fate's Ultimate, and Recall.