What is the etymology of the suffix(?)たがい in 仲たがい and how is it used to express discord? If this is a suffix as such, is it used in combination with any other words?
The たがい in 仲たがい is another, rarer pronunciation of 違い (usually pronounced ちがい), a noun meaning "difference". This is not to be confused with 互い, which has the same pronunciation. 仲 has the general meaning of "relationship", and is the same character from 仲良し ("close friend").
So overall we have a compound noun:
仲 - relationship
違い - difference
meaning differences (ie disagreements) in a relationship causing discord.
What is the etymology of the suffix(?)たがい in 仲たがい
The etymology is tagapi > tagaɸi > tagawi > tagai. It is the nominal form of the verb taga-u (違う). As such, suffix is not the appropriate word. As a verb, there was the expression 仲を違う (naka wo tagau). This is the source of the compound 仲違い (nakatagai). This taga-u and tiga-u (or chiga-u if you like) are related. taga-u is the older verb and is essentially replaced by tiga-u.
and how is it used to express discord?
ジョン answers this sufficiently.
If this is a suffix as such, is it used in combination with any other words?
There are other compounds which include tagai. They include:
- 人違い (hitotagai)
- 門違い (kadotagai)
- 片違い (katatagai)
- 方違い (katatagai)
- 聞き違い (kikitagai)
- 心違い (kokorotagai)
- 思い違い (omoitagai)
All of these are old and unlikely to used much in modern Japanese. If you are into history, there is the 角違一揆 (kadotagai ikki) riot of 1346 in Bungo.