Were there in the USSR monasteries decorated with Soviet decorations, particularly, the Order of the Red Banner?
No religious institution has ever been decorated by the Soviet government. I do not have a definitive proof (that would require searching through all the decoration records, which, given the fantastic generosity in decorations - tens of millions decorations awarded - is beyond my capabilities), but I am quite confident that this has never happened, even though I can understand the doubts someone who never lived in the USSR might have.
When thinking about USSR and its ideology, one has to remember the motto "We do not penny-pinch on ideology" - "На идеологии мы не экономим". Ideological concepts were taken quite seriously - only the need for the atomic bomb saved quantum mechanics, and, despite chronic food shortages, genetics was still condemned.
Until 1943, all religious institutions and their "officers" were treated as enemies and persecuted at any opportunity.
The official phraseology underwent a major shift from "internationalism" to "patriotism" during the the WW2, including letting the church elect a patriarch and accepting donations from it, as well as decorating some individual priests.
However, the idea of decorating a church institution still is preposterous.
The reason is that the Soviet ideology invested organizations and institutions with a certain degree of sanctity (this is actually the reason why Kamenev/Zinoviev/Buharin &c were not exonerated in 1950-ies together with lesser victims - because they were condemned not just by the court, which could err, but also by the party - which is infallible).
Similarly, decorating an individual priest says "even though he is confused and brainwashed by a retrograde ideology, he did the right thing and we recognize that".
However, decorating a church institution legitimizes it ideologically, so it is completely unacceptable to the Soviet officialdom.
Therefore, even if a monastery did something good in the eyes of the Soviet government, they would never decorate it, they would decorate the individuals responsible (e.g., monks). An example is the decoration of Alexy I with the Order of the Red Banner of Labour for his participation in the attack on the Ukrainian Greek Catholic Church in 1946 - the person was decorated, not the institution he lead.
I think what you heard ("Order of the Red Banner XXX monastery" - "краснознамённый монастырь") was a joke.
This was actually a very common theme in the Soviet folklore, playing on the officialdom's frequent abuse the "organization decorations", e.g., "Ленинградский ордена Ленина метрополитен имени В. И. Ленина" (Leningrad's of order of Lenin subway named after Lenin).
It was used to imply that the entity in question was important and well known (as well as "recognized by the government as legitimate"). E.g., one could say "московская трижды краснознамённая хоральная синагога" ("moscow choral synagogue thrice decorated by the red banner order") to emphasize its official standing with the soviet bureaucracy, as opposed to a group of Jews (secretly) praying together in a random basement.
Another common joke was referring to arbitrary cities as hero-city (e.g., город-герой Нью-Йорк).
No, they were not. Church was an "opium for the folk" so church institutions could not deserve to be decorated with respected symbols of communism. This Marx phrase was the official view.
Soviet regime was strong enough on its own. There was no need for it to accept any help from religious institutions, even if offered. And the official negative position was firm enough so that voluntary decoration with such symbols was unlikely to be helpful.