One of my favorite math teachers in high school told us that whenever we got stumped, we should draw a picture to get a better definition of what the problem was.
Here's a recent (12 June 2015) article by Louise Coakley, X-DNA's helpful inheritance patterns. Coakley published several graphs with the possible lines of descent marked out.
With no X inherited from some ancestors, varying amounts inherited
from others, the randomness of DNA recombination at each generation,
and occasional sticky segments passed down intact over several
generations, X-DNA can be quite unpredictable and difficult to
interpret exactly where it came from.
Before you try to sort out where the match could have come from, take a moment to think about how the X gets passed down for males vs. females.
Because males only inherit an X chromosome from their mothers, if a
male has an X-match in his DNA results, the shared ancestor must be on
an ancestral line that follows the male X-inheritance pattern
Because females inherit X chromosomes from both parents, if a female
has an X-match in her DNA results, the shared ancestor must be on an
ancestral line that follows the female X-inheritance pattern
Below each of these statements she has a chart where the pattern is marked out in green. That shows each of you where your own Xs could have come from. It is not easy to work any of these problems out when you are only comparing a match between two people, and it is easier to work out where the X could have come from if you had a match on it.
For your second case, with the cousin who has an X-DNA match but no autosomal, Coakley says:
It is common to share segments of X-DNA with people who share no
significant amount of autosomal DNA.
The situation is complicated further because men inherit the X from their mothers unchanged, but with women, they get an X from their mom which has DNA recombined from both their mom's X chromosomes. If this gives you a headache just to think about it, you aren't alone.
Coakley has a section in the article Practical uses of X-DNA where she goes over different scenarios and how the X-DNA might pass down.
She has links to download the entire article in PDF format, high-resolution versions of the charts, Excel templates to help you map out matches, and more links to other articles that talk about X-DNA.
Sometimes the explanation of a topic that makes it clear for one group of students doesn't work for everyone. Instead of recognizing that the teacher hasn't made the topic clear yet, the students who didn't "get it" on the first try internalize this mis-match and say it is because they are dumb.
The real issue is that some subjects are difficult, and we aren't always fortunate enough to discover the explanation that works the first time we try. The students who are forced to take a second and third look often end up with a better understanding of the topic in the long run, because they've had to think about it more. If this article doesn't make it clear, follow her other links and read more articles. Give yourself some time to let things sink in, and don't be intimidated if you have to read articles multiple times before you feel like you 'get it'. My experience is that even when I think I've understood something the first time out, it can take six months or more before the information really sinks in.