This a very restricted use of がち.
Typical がち meaning
Your basic understanding of がち as a "接尾語" is correct: it indicates a tendency or something happening often, and can be constructed with a host of base words, as in 病気がち, 遅れがち, etc.
It can, grammatically speaking, and in this meaning, be suffixed to any word. Which ones are acceptable or not is only a question of usage: for example, 疲れがち is grammatically correct but in usage 疲れ気味 is usually preferred.
がちに special case
Now, as you mention, 遠慮がち（に） doesn't really fit with that interpretation. It doesn't have the meaning of 「遠慮が多い」 or 「遠慮する傾向」 (遠慮しがち does instead).
It has the meaning of "as if" instead (here: "as if restraining oneself"). In japanese: 「遠慮しているような感じ」 (taken from the referenced site below).
Contrary to the 1st, typical meaning above though, in this meaning it is highly idiomatic and can't really be constructed grammatically by suffixing any noun. I gathered 伏し目がち（に） and ためらいがち（に） falling into this category, but I doubt there are many more.
In these cases, the がち construct seems to have the function of filling a semantical hole no other established construct can fill: it expresses a link between two semmingly incompatible actions. In the following examples, at the face value of the word meanings, it should either be one or the other: 「遠慮する」 or 「言う」, 「伏し目になる」 or 「相手を見る」. Not both together. Hence, when these actions are associated together using がちに, its possible translation of "as if".
彼女は遠慮がちに言った
彼女は伏し目がちに相手を見た
怒りがちに ?
Finally, to answer your last question:
- 「怒りがちだ」 exists and is correct, but fits the first, typical meaning of
がち (tendency)
- 「怒りがちにドアを開けた」 sounds unlikely to me, as the 2 actions are in no way seemingly incompatible ; something like 「怒りがちに無言になった」 maybe, we could consider. However:
- 「怒りがちに」 is simply not established in usage, so it makes it incorrect, and likely to confuse the person you're speaking to. Furthermore:
- 「怒りっぽく」 exists and fits the meaning already, and
- To say "he opened the door angrily" 「怒ってドアを開けた」 is sufficient.
Thinking of the restricted few established uses of がちに as having the meaning of "as if" may further help understanding why it wouldn't be warranted in this example.
Reference: 接尾語「〜がち」について分析
