Comprometerse in this context = to commit in English (afraid to commit/afraid of commitment).
Llegar a un acuerdo = to compromise
No I don't think so. Going out/dating/seeing each other all refer to the early stages of a relationship (I would say the first six months of a relationship, but other people's opinions may differ). But certainly, there isn't any real commitment in any of those types of relationship yet, by either party.
Long-term committed relationship is much, much higher in the commitment stakes than any of those you mentioned. For a start, long-term = over many years.
Some people don't want to get married, so for them 'a long-term committed relationship' is as committed as you can get. Basically the same as a marriage but without the sheet of paper. It is a high level of commitment.
Maybe it's a less common term because people who have been in a relationship for several years tend to be married? I'm not so sure it's do with a fear of commitment, although it's an interesting idea.