Firstly, no problem. I enjoy helping people out. When I first started learning, I used to post on the French and Spanish forums, but now I've forgotten most of those languages =/. Also, when I first started, posting on forums helped me to learn more. Even if I didn't know the answer, I taught myself whatever the person was asking about, so that I could respond.
Secondly, I was really wondering whether I not I should have used "taberun deshita" as an example.
The reason I used it was because it's a case where the "~n/no" is necessary. It might not be one of the "~te imasu" or "~te irun desu" patterns, but think about how the "~te irun desu" pattern is just an extension of the "dictionary form + n/no desu", but with "iru".
This usage is not to be confused with "o sagashi desu" or "stem + desu". In this case, there's not really a nuance. Stem forms of verbs are used as nouns. Taro could have just as easily said "Apaato wo sagashite irun desu ka?". His deciding to use the stem form seems kind of arbitrary. I don't think there's any specific case where using the stem as a noun is required grammatically (since gerunds/nouns can be formed with "dictionary + koto", just as easily).
So basically, using "taberu" as an example,
"taberu (dictionary) no desu/deshita." or "taberun desu/deshita" "I eat/ate", implying an explanation for a completed action in the present or past.
"tabete iru no desu/deshita" or "tabeterun desu/deshita" "I am/was eating", also implying an explanation for an ongoing action in the present/past.
While "tabete imasu" just means "I'm eating".
I've never seen the stem form of "taberu" used specifically as a noun (probably because, as you say, it sounds awkward), so instead I'll use "hanasu".
"Hanashi desu" "It's a story".
So often using the stem as a noun has different connotations, to say, just the gerund of a verb.
"Sakana wa, doko ni mitsukatta no?" "Where did you find the fish?"
"Sono kaeri ni, ike de asatte irun deshita. Soshite, sakana ga gohiki wo te ni ireta"
"When returning/On the way back (lit. in that return), I was fishing in the pond (the ~te irun desu construction implies explanation), and I caught 5 fish! (literally, 5 fish entered my hand)".
"Sagashi desu" "It's a search". So Taro says "Is it a search?" "O sagashi desu ka?"
Beyond this, however, I'm not really sure what to say. I don't think there's a nuance to its usage, but I'm not too sure.