There is quite a bit of variation with the slender d, but this is pretty standard when not at the beginning of a word, so I would pronounce idir pretty well like teacher - with no t and the r stronger than that found in southern England and weaker than in Scots. That is pretty well what you wrote, except, of course, for the r which is completely different. This is a dialect variation for slender r found mostly in Lewis and is indeed like a weak th.
Says who? Because of the enormous bias in favour of certain languages in the past – see https://forum.duolingo.com/comment/35456912?comment_id=39104218 – we have to be very suspicious of any claim that a Gaelic word comes from English unless we have evidence of recent and/or thorough research. That means that this claim – as indeed with any claim for an etymology – is worth little without citing sources.
In this case I guess it came from MacBain (1911 from first ed. 1896) as there aren't any other sources I know of for words that do not go via another language. It may have come via Wiktionary as that has acknowledged taking it from MacBain and says exactly the same thing.
What is interesting about this word in MacBain is that he usually makes an effort to find similar words in a variety of relevant languages. But in this case he seems to have missed the Old Irish word brothchán which means 'broth, pottage, soup, gruel'. This oversight is so blatant that we must assume he did not do any proper research and simply presumed it was English.
There are quite a few quotes from the Old Irish period, so if the Old Irish word comes from any kind of English (which is very rare anyway) then it is from Old English broþ. It seems more likely that it came via Late Latin brodium from Frankish from Proto-West Germanic *broþ.
If the Gaelic did come from Old English or Middle English, or even the Old Irish, then we would expect the word to be *broth, since th was pronounced the same in all these languages at the time. The form brot matches the Latin much more closely than it does the English or the Old Irish.
So overall I am not convinced it came via English, although it is clearly Germanic in origin. It seems much more likely that it comes via Latin, with the Old Irish form being influenced by Frankish etc.
(Note that þ is the way that th as in broth was written in the older Germanic languages. The letter is called 'thorn'.)
Don't try to pronounce both ls separately as toigh leam is the historically correct spelling. Pronounce the toigh as toy in English.
Leam has very different pronunciation according to dialect. I would say the whole thing like toil 'em in English, or toy Liam if speaking slowly and carefully, but some of the speakers on the recordings seem to be saying toy lume, with the lume rhyming with spume so it has a y-sound at the beginning of the vowel.