The present habitual ...mar bím tuirseach should be used here. Tá mé tuirseach is a simple statement of fact at the present moment, but codlaím is a habitual action. Imagine codlaím was qualified temporally and the sentence was, Codlaím gach oíche mar tá mé tuirseach. This sentence would clearly be nonsense because it's translation is "I sleep every night because I'm tired (right now)." Bím tuirseach fits in much better because it's meaning is roughly "I'm tired (every night)." This remains true even if there's no explicit temporal adverb, because of the habitual aspect inherent in codlaím and bím but absent from tá mé
so, Taim means I am ,when it is the subject of the sentence ? and here, it is TA ME because( i think) the meaning is ( more or less) : to me the tiredness ( if tiredness exists in English) I mean :the construction of the sentence is : the fact of being tired is mine. If it is not so, then i don't understand the TA ME instead of TAIM. Gaelic is really an interesting language which requires a different way of thinking , especially for those of us who speak Romance languages.
There is no difference between "tá mé" and "táim", I assume the development of the latter, where the pronoun is subsumed into the verb, allowed for efficiency of speech.
'tuirseach' is an adjective (tired). Having said that, I think the far more common expression used to complete this sentence would be, ...mar tá tuirse orm. (Tiredness is on me). The irish adjective 'tuirseach' is more commonly used to modify a noun - he is a tired/weary man.
tá tuirse orm is not in any way "far more common" than tá mé tuirseach, especially when tuirseach is combined with intensifiers like tuirseach traochta.
While many states that use a predicative adjective in English are expressed using a noun and a preposition in Irish, tuirseach is one example where the predicative adjective construction is very widely used.
"Táim" is exactly the same as "tá mé"; Táim is called the "synthetic" form of tá mé. The default word order of Irish is Verb-Subject-Object, so you can safely assume that if there is a pronoun directly after the verb, that pronoun is the subject of the sentence.
I think so too which is why I wish they would give literal translations as well as English equivalents. I think when you know the literal translation, you will eventually think that way, especially with prepositions, and it will help you know when to use that preposition in other instances...just my thoughts anyway, and now there seems to no one here to constantly argue with me so I get to express them.
So, I totally get your frustration. But, as a native, American English speaker,(with a knack for languages), but, unlike French, Spanish, and German, (which I find relatively easy) Gaelic is a little more difficult. (Ha! A lot more difficult!) The syntax is hard to translate. I decided to hit it head-on like a child learning a language. Don't overthink it. Repeat, repeat, repeat. Which, is how it seems, Duolingo teaches. Frankly, the best thing I ever did was to stop trying to translate literally(based on English, German, or "romance languages ".