Firstly let me state about your second question as a brief answer. As you inquired:
“Is it a man who goes into seeking a relationship in Mutah or the act
of seeking is valid from both directions?”
I didn’t find any specific related issue to that, hence, at least it could be said that it is not considered as a haram practice as much as I made research. But perhaps it could be nicer for women to endeavor not to take a step at the first side. (Or perhaps it is not true) / Wallah o A’lamo (And Allah knows better)
But regarding your initial main inquiry, seemingly it is okay, but there should be paid attention to some items regarding Mut’ah which ought or ought not to do, such as:
- Excess in doing it that it can be considered as sensuality
- Being hidden the issue from the family …
- Keeping the respect of the permanent wife (and permanent marriage)
- If it is below his dignity to do this (which is harmful for his
personality or responsibility …
- Exposing to accusation (e.g. if some people are aware about that,
then they’ll accuse you as Zina if they are not aware if you have
done it as Mut’ah …
- Doing it with the girls who are virgin (& it will destroy
the opportunity of permanent marriage)
- And so forth.
Consequently, doing Mut'ah for a married person would be permissible, but s/he ought to note some points as I termed (Although others ought to note them as well)
Those above-mentioned items can be counted as some remarkable items which I endeavor to state them briefly and approximately. For more complete matters, you’d better refer to the reference below.
Reference: