How do various superheroes do their part both on superteams and in their own neighborhoods? That is, how do they split their time?
Not sure I understand the question. As overexposed as Batman and Wolverine are, you could read all of their adventures published this month in all the titles they appear in in a little under two hours. One month minus two hours is a lot of time unaccounted for, hardly less so when you consider that Wolverine has unlimited access to a Lockheed SR-71 Blackbird, among the fastest planes in the world, while Batman personally owns numerous aircraft and a JLA teleport tube. These guys have time to kill and then some.
Spider-Man, another really overexposed character, has a salary and budget on a par with my own. Luckily, most of his adventures take place within an eight block area. When he joins the Avengers formally later this year, make it a ten block area, plus the Avengers also own very fast aircraft.
Well, Clark Kent and Peter Parker are collaborating on an exposé for Rolling Stone, Clark of course writing the story while Peter handles the photos illustrating it… 
(But doesn’t that sound like the perfect lead-in for a crossover story?)
Teen Titans is a weekend thing for the younger members - During the week, they work in Smallville, Keystone, Gotham/Bludhaven and wherever Wonder Girl lives, on the weekends they fly/run out to San Fransisco and hang in Titans Tower. Speedy will no doubt be the same when she joins - Star during the week, SF on the weekend. The other members presumably stay in San Francisco, so any ‘solo’ adventures they have would be there, just like team stories.
The JLA doesn’t make it quite as clear how the various members budget their time, but it’s clear that most of them spend most of their time in their home cities, but are usually available to work with the League - although it’s implied they do live on the Watchtower sometimes.
Others, I can’t really comment on.
Justice Leaguers are “on call” with someone on monitor duty at all times, and they have a regularly scheduled weekly meeting.
Avengers actually live at Avengers Mansion, unless they have some compelling reason not to. Iron Man lives elsewhere, but can be gotten on short notice. Captain America has a residence at the Mansion and another somewhere else and splits his time between the two, but usually there are at least some members who take up regular residence at the mansion (Hawkeye, Vision and Scarlet Witch lived there full-time until recent events; I suspect Sentry and Spider-Woman will be resident Avengers pretty soon). Some Avengers are full-time, most are not.
With the Teen Titans, again some members live there full-time (Cyborg, Starfire, Beast Boy) while others commute in on weekends or as needed. This arrangement made a lot more sense when they all lived in or near New York. Robin has a pretty rough commute.