During Iddat women not allowed to even talk with some "Na Mehram", Even she is not allowed to see "Na Mehram".Then how can a women can be committed to someone or get engaged with someone.So this is not allowed during the time period of Iddat ,yes it is allowed after Iddat duration.
Your First Question:
Which schools of thought allow/disallow it?(from the 4 major ones)
The four schools: Hanafi, Maliki, Shafi'i, and Hanbali, allow Katb-Al-Kitab (marriage contract) as a must rather than an optional choice. Allah Almighty described it as a solemn covenant in Surah Al-Nisa (4) — Verse 20 and 21:
In Arabic (the exact wording):
A temporary marriage in Islam is called nikah al-Mut'a or al-Mut'ah and was a common practice in Jahiliyyah.
There's a difference of opinion among the different Muslim sects about it:
The twelver Shi'a say that nikah al-Mut'a or a temporary nikah is allowed and a sunnah that one can do it endlessly beyond the four wives, but it is better to avoid it if one ...