I know

Sláinte
Sláinte mhaith (this old me an Scottish guy in 1996 when I was in Shropshire, England for an apprenticeship exchange)

Sláinte is táinte (this is translated as Health AND Wealth)...but I think it is not correct because "agus" is "and", or am I wrong?
Sláinte agus saol agat (Health and a long life)

So I created this one by myself:

Sláinte mhaith agus saol agat. (Good health and a long life) Would that be correct to say in this form? Or is that too muc or grammatically incorrect?

Yes, you are.
"is" is a very common abbreviation for "agus"



It's only a life.
"Get a life" isn't very polite
a long life is saol fada



I'd say
... agus saol fada chugat
chugat = to you
but agat might work as well. (in many dialects agat replaces chugat anyway)

Omg...
Good I didn't use it.





So....it'd be
"Sláinte mhaith agus saol fada agaibh" to more people or "Sláinte mhaith agus saol fada chugaibh"?

Grma...

This mistakse is the best way to show my difficulties keeping that stuff in my mind.

And why do we use a form of "chuig = toward" and not from "do = for/to" in this case duit?

Changes in semantics over the generations, dialect differences...

There would seem to be more of a sense of 'mental movement' (intent) to 'do' than to the transportive sense of 'chuig' (or maybe it's just me), but as has been said, 'chuig' reduces to 'eig' so you get a confusion and overlapping with 'agat' over time

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The use of agat/agaibh is based on the way one says "to have" in Irish (and Scottish Gaelic). For example, Tá teach agam means "I have a house" (literally: "there is a house at me"). So, when you say Sláinte mhaith agus saol fada agat/agaibh!, you're actually saying "Have good health and a long life!".

Saying Sláinte mhaith agus saol fada chugat/chugaibh could be seen as meaning "Good health and a long life to you!", although, as noted, the choice is really more of a dialectical thing. Note that you can say it both ways in English, as well (as shown by the translations I've given).

While the word "do" can mean "to" in some cases, it can have a number of other meanings as a preposition, and if you were to use it in this case, Sláinte mhaith agus saol fada duit/daoibh would come across more like "Good health and a long life for you!". That isn't illogical, but it would sound a bit off, just as the English translation does (while also logical).

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.