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?
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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:
meaning differences (ie disagreements) in a relationship causing discord.
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.
ジョン answers this sufficiently.
There are other compounds which include tagai. They include:
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.