To answer I will address each issue raised individually.
I wouldn't say that Torah-observant Jews are ever "compromising" in Jewish Law.
The Rebbe was a Shulchan Aruch Yid and always followed Halacha to the detail. (This is obviously true of every Tzadik. Following Kabbalah or Chassidus does not negate that). That said, Halacha recognizes that in cases of great need, such as saving a Jewish life, different opinions can be relied upon, not rejected altogether.
"not being able to daven with a minyan"
I don't think Davening with a minyan has the same status as Lulav and Esrog which is explicitly in Torah.
"not having meat for yom tov, or perhaps not having a lulav and etrog
on sukkot etc."
These are often easily obtainable. Even Cholov Yisrael milk has become accessible though I know many Shluchim who froze milk, or even used dehydrated milk. I don't think there is a Shliach who ever went Sukkos without a Lulav and Esrog.
A major issue that is absolutely uncompromising is Mikvah. The Rebbe was adamant that Shluchim, regardless of where in the world they are, must have access of some sort to a Mikvah (i.e. flying to get there, having access to an ocean, etc.)
"how is a person allowed to go on shlichus?"
If you ask any Shliach this question they will most probably answer you "with כח המשלח (the strength of one who sent him)". The Lubavitcher Rebbe took an enormous responsibility upon himself in making that decision, and many critics at the time saw the program as spiritual suicide.
The Rebbe often reference the Halachic ruling in Kashrus that "איידי דטרידי סימנין לאפוקי דם לא בלעי" (Gemara, Chullin, 8:2), that when something is extracting it does not absorb. So too, by being focused on influencing their communities they are not influenced by them.
In hindsight, we obviously see that tens of thousands of families were transformed positively through the Rebbe's work.
"Is shlichus permanent?"
Usually, when Shluchim agree to go out, it's a one-way ticket. The Shlichus finishes when Moshiach comes, which the Rebbe ensured us would be in this generation.
"Can a person who has agreed to go on shlichus
ever leave that post?"
For various reasons, there are some Shluchim who do leave their post, or move to another place of Shlichus.