Let's say someone would have tefillin on him during mincha shemona esrei. Would he get a mitzvah for doing so? Would that be allowed?
The Shulchan Arukh (OC 37:2, echoing the Tur ibid.) writes:
מצותן להיות עליו כל היום אבל מפני שצריכים גוף נקי שלא יפיח בהם ושלא יסיח דעתו מהם ואין כל אדם יכול ליזהר בהם נהגו שלא להניחם כל היום ומכל מקום צריך כל אדם ליזהר בהם להיותם עליו בשעת קריאת שמע ותפלה:
The commandment is to have [the Tefillin] on all day, but because they a clean body that he won't flatulate while [wearing] them and he won't be distracted from them and not every man can be careful about them, they are accustomed to not wear them all day, but nevertheless every man must be careful to have [the Tefillin] on him at the time of the reading of Shema and Tefillah.
Basically: every minute is a Mitzva and it's most important to wear Tefillin during Shema and Tefillah.
If you have to choose, it would seem Shacharit is more important than Mincha for reasons like you get Shema as well as Tefillah and we generally take the first opportunity to do Mitzvot ("Zerizin Makdimin"). Plus, people are out and about in the afternoon usually so it's not easy to have Tefillin around.
In general nowadays most communities do not don Tefillin at Mincha, and doing so in those places could constitute Yuhara (see Igrot Moshe OC 4:34), yet there have been many individuals over the years who have been careful to don Tefillin at Mincha, at least in private (see eg. Magen Avraham OC 37, B'Yitzchak Yikarei ibid., Ma'aseh Rav 63, Ben Ish Chai I Vayaqhel 6, etc.). Some are particularly careful about this on fast days, to help make up for the lost blessings on food when working towards 100 blessings (Beit Yosef 46).
Yes, it is allowed, and yes, one can fulfil the obligation of tefillin by wearing them during mincha – or indeed any other time of the day.
I have often seen, especially chassidim, wear tefillin at mincha, either because they wanted to wear their Rabbeinu Tam tefillin during a prayer or because they for some reason (e.g. travel or Tisha B'Av mourning) were prevented from wearing their regular (Rashi) tefillin during shacharis.
From what I remember, TANAIM, EMORAIM etc. would wear it all day long.
Since we are not at their level, we only wear it for Shacharit.
I'm sure that if that is the first time that day, that would count as putting it on for that day. Maybe not the preferred method, but still better than not at all.
Yes, there are groups who wear tallit and tefillin at mincha.
These include Jews who follow the teachings of the Arizal who specifically says to wear tallit and tefillin for mincha, and on a Tallis on Shabbat (and Yom Tov) mincha as well. This is also brougth down by the Ben Ish Chai in parashat Chayei Sara.
In these circles they normally wear Shimusha Rabba tefillin or Rabbeinu Tam ones in their absence. Some chassidic background individuals do make a point of wearing Rabbeinu Tam tefillin at mincha to say kedusha with them on.
As per raw halacha, see this shiur by Rabbi Avi Grossman discussing how according to the Rambam, Bet Yosef, Aruch HaShulchan, and even Mishnah Berura one needs to wear a tallit gadol for all the prayers, including mincha. See Kitzur Shulchan Aruch 9:1 that every Jew should endeavor to wear a tallit gadol during prayer including Mincha.
See also M”A 37:2 in name of Menahem Azariah da Fano that he instituted to wear the Tefillin also by Mincha (See Beer Heitiv, Eliyah Rabba, and Shaarey Teshuvah 37 ad loc; Reishis Chochmah 6:54 and 63; Biur Halacha 37 “Beshaaas”).
See Rav Moshe Feinstein zt"l in Igrot Mosheh 4:34 that is ok to do so if you have adopted this custom. Obviously, if it is not your minhag to wear tefillin at mincha, it is ok to wear them at mincha if you had no chance to wear tefillin on that day until that time.
Nevertheless, tefillin are not worn in that case on Erev Shabbos or Yom Tov (Shaarey Teshuvah 34:6; Biur Halacha ibid; Kaf Hachaim 37:11) unless you have not done so previously and this is the last chance to do the mitzvah. If this is the case, it would be advisable to wear them privately first and then join the minyan if possible.