Can anyone unravel for me what seems to be a mix-up between several verbs?

Understand = tuigim = tigim in Connacht.

Dinneen's has come, arrive = tigim = tagaim = teagaim, verbal noun = teacht

Then Dinneen's has this comment under the entry for teacht: "originally with dul, vl. of tiagu (I go), tiachtain (tíochtaini) was the vl. of tigim (I come), tuidheacht of do-tiag (I come), tetacht of do-in-tíag (I come); from the latter prob. comes the mod. mg. of teacht (var. tocht)."

Er, does this mean teacht was original the verbal noun of dul? And different dialects mix up "come" with "understand"?

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Er, does this mean teacht was original the verbal noun of dul?

Yes, but that was a LONNNG time ago, i ré na Sean-Ghaeilge.

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And different dialects mix up "come" with "understand"?

No, not really. It's kept sorted out in most people's speech, with "tuigim" in most dialects and the Standard for "I understand", and distinct forms for "I come", such as "tagaim, teagaim". The only apparent confusion is comes when you compare across dialects, and even then it's less confusing than our American English confusions of "writer" and "rider", or "ten" and "tin" (widespread in Southern states).