There are runners on 2nd and 1st base, and they're both leading off. The pitcher throws to 1st to try to get him out - is the runner on 2nd base allowed to run to 3rd?
In baseball (at least as far as the MLB rules go; other levels can have different rulesets, though they tend to be similar), the ball is alive except when it is specifically defined as "dead". That definition is in Rule 5.09 in the current rulebook; while that's a longer rule than I want to quote here, what you'll find is that there is no reference to a pickoff causing a dead ball situation.
The ball can become dead when a runner calls for time, and the umpire grants it:
5.01 (b) (5.02) After the umpire calls “Play” the ball is alive and in play and remains alive and in play until for legal cause, or at the umpire’s call of “Time” suspending play, the ball becomes dead.
After a pickoff it is certainly possible that the runner might ask for time and the umpire might grant it, but if the runner on second or third was in the process of attempting to advance, the umpire should not grant it.
5.12 (8) Except in the cases stated in paragraphs (2) and (3)(A) of this rule, no umpire shall call “Time” while a play is in progress.
(2/3A are referring to specific problems with lighting or injuries, and not relevant here.)
As such, it's permissible for a runner on second or third to attempt to advance during a pickoff play, assuming the pickoff play itself was during a live ball situation (otherwise it's pointless, as runners are not liable to be put out during dead ball situations). It isn't a common occurrence at the major league level, but it is certainly possible in the case of a double steal (where both runners are attempting to advance, and the pitcher guesses correctly and tries to pick one off). At lower levels, it may be more common as pickoff plays and subsequent throws carry more risk of overthrows or players not paying sufficient attention.
The short answer is yes. But, in practice, it typically doesn’t happen because of the risk of being thrown out. Assuming the ball is currently in play, there’s nothing stopping a runner from trying to advance to the next base. In the situation you’ve described, the usual course of action for the runner on second is to retreat part way to second base on the off chance that the throw to first is mishandled in some way. Returning part way would allow the runner to advance much easier if the throw to first was errant.
Basically, as long as the ball is "in play," a runner is allowed to run to any open base. The runner from first is allowed to run to second, or back to first. The runner from second is allowed to run to third, or back to second (in the last case, he prevents the runner from first from using second base).
Most runners that are "caught" in this way prefer to run to the nearest base. If the runner from second is actually closer to third base, he's past the "point of no return" and should run to third base, and leave second base open to the other runner.
What usually happens is that the defense concentrates on one runner, leaving the other runner free. The most likely scenario is that they concentrate on getting out the first base runner, which means that the second base runner gets a "free pass" to third. If the defense then throws to third, maybe they overthrow and both runners advance. A good throw to third might nail the lead runner, but the first base runner advances to second in compensation.
The most likely result in these situations is that one runner is sacrificed to allow the other runner to advance. If both runners manage to advance, that's a good result for the batting team; getting both runners out is a good result for the defense.