A fielder may use the colored base when the put out attempt is coming from foul ground. The reasoning is that if the fielder is in foul ground and you require her to still use the white base, it would force the fielder to cross over on a collision course directly into the runner's path. The "safety base" wouldn't be so darn safe in that case!
For the question in the first post, I would have an out.
There are a few exceptions where the fielder may use the colored base. There is the one mentioned above, when the fielder is completely in foul ground and must come back to the base. As someone mentioned, another is when an errant throw pulls the fielder off the white base and into foul ground, or onto the colored base. The last one is when the throw to first base is coming from the foul side of the first base line, like after an uncaught third strike.
Whenever the defense uses the colored base, the runner may use either the colored or the white portion. Most of the time, the runner will be heading directly to the colored base on a routine play. If something happens that causes the defense to use the colored base, it would be unreasonable to expect the runner to recognize that and require them to go to the white base at the last second. So, they can go to the white if they want to, but the rule doesn't require it.
After the initial put out attempt on the batter-runner, the double first base becomes one big base and either side can use either half of it. Runners can tag up or lead off from either half, and fielders can use either half when putting out a runner who left early on a caught fly ball.
And, of course, if there is no initial play at first (for instance, on a double to the outfield) the runner can use either half when rounding first base.