Does it look bad to cite a single review paper repeatedly for many different things?

First of all you should cite original sources. The review paper can be cited for whatever conclusions or original work stem from it, not for whatever sources it, in turn cites. In your example, citing the same sources over and over again in a single paragraph is unnecessary. You are better of starting the paragraphs stating that the information is sourced from [1] in some way.

In general, citing a single source can be reasonable if it is a really original work and the only possible source for much information. This is, however, rarely the case and so my thought is that regardless of the situation, most will see it as somewhat suspicious. It all depends on whether the paper really is the only source for the information.

So, general advice is: avoid doing what you outline, unless it is the only possible way (from the point of view that no other sources exist.

Despite having constraints on length, you should be able to fit a reasonable number of references in. I am sure that you may have to really think twice about your text. I have certainly been in situations where I have has to ask myself if each end every sentence written is () necessary and (b) concise enough.

For (real) reason of space you can surely cite a review paper or a book, but, to reduce the number of citations, I would do it in a different way with respect to your example.

For instance:

Common sorting algorithms are the bubble, the merge and the quick sort (see [1] and references therein for more details).

or

Common sorting algorithms are the bubble, the merge and the quick sort (see [1] for a review of their properties).