In running clock games, the clock runs during free throws (except, perhaps, in the last two minutes or so, depending on your rules), so if they foul, then request and are granted a timeout, when they come out of the timeout to shoot the free throws, the clock starts again. That's because once the timeout is over, the clock starts when the ball becomes live. This is different than in a stop clock game
It depends on the league/tournament. In the Labor Day Tournament I have been working at the past several years, the clock is running clock except for the final 2 minutes of the second half and any overtime (unless it's a 20 point or more margin).
In the first half of these games, teams that actually know strategy will call time out in the final minute to stop the clock during fouls so that the clock won't stop until after both free throws are shot. Perfectly "legal" under this tournament's rules.
On the other side of the coin, I've been at a summer lague where calling a time out at any time prior to under a minute to go in the 2nd half does not stop the clock. (The perfect way to keep coaches from spending too much time in a timeout
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It's all a matter of how the tournament implements its altered timing.