1. "To keep up with" and "to keep pace with" can mean the exact same thing depending on the context. For example, in track and field, another sprinter can "keep up with" and "keep pace with" Usain Bolt. It means whatever or whoever is doing something is able to maintain a certain level of activity or output and usually when comparing something or someone to another. If I am running in a race, I can "keep up with" the leader and "keep pace with" the leader. It means the same thing. If I am climbing a mountain with an experienced climber and I am doing a good job, I am keeping up with the experienced climber/keeping pace with the experienced climber. Again, it means the same thing. There are only a few exceptions to this general rule I can think of (for example, the American English idiom "to keep up with the Joneses." Only "to keep up with" works with this idiom). This has an interesting cultural meaning and don't worry about these few exceptions for now. 2. "to come up" sometimes is referring to an event or some obligation. For example, if someone needs to make an excuse to not go on a date with someone, they may say, "I'm sorry, I can't go on a date with you tonight. Something really important just came up that I must do." It usually implies a sudden change, as in, "something really important just came up, I cannot go to the meeting today." It is similar to "to come about" in another context, which is to approach something. For example, you can be on a nature hike in Africa and suddenly "come up to" or "come about" a lion. To come up to something is more frequently used than to come about something. Now, to make things even more confusing, you can also "happen upon" something/someone. For example, you could also "happen upon" a lion during your African safari. This means the exact same thing! In general, "to happen" is most often used where you could also use "to occur" or "to take place."