What you're using here is the imperative mood and the expected forms for that would be téimis or téadh muid, but GnaG says that téimid is acceptable as an alternative form: "Often, in the 1st peson plural, also the indicative-present tense suffix -imid is used: Téimid = let us go." (http://nualeargais.ie/gnag/zeitform.htm#Imperativ)
Which is where I would disagree with you, as the verb téigh doesn't "end in a slender sound" - it ends in a slender consonant.
The problem with trying to follow the Tips & Notes for Present 1 is that they say that, for 1st conjugation verbs of 1 syllable, "the ending is, generally, added directly onto the stem", and "These verbs have only one syllable¹, and the root form seen in the dictionary is identical to the stem used for verb conjugation", which would mean that the 1st person present tense form of téigh would be téighim and the 2nd person present tense would be téigheann.
The problem here is that this doesn't take account of single syllable verbs that end in -igh, and it turns out that 1st conjugation verbs like téigh actually take a broad ending in the present tense - téann, even though the "root" form has a slender ending.
The page on Gramadach na Gaeilge that documents this is here.
I think you have a few concepts mixed up here.
"h" is not really a letter more often than not in Irish. So where you see "gh" the "h" is replacing a diacritic indicating lenition. In the old Irish typeset an Clo Gealach it looked like this: "ġ". (téiġ) I like to describe lenition as pushing air through a consonant sound, but that's probably lazy. But I have found it surprisingly effective in trying to get better pronunciation than my crappy Dublin school Irish yielded.
Broad and Slender are a totally different concept. The slender vowels modify the pronunciation of the adjacent consonant in a different way . I like to lazily describe this one as trying to pronounce the consonant closer to your teeth with your tongue pushed up. G is a good example here because it's has close corollaries in English so broad G is G in Goat. Slender G is G in Git.
So I disagree with SatharnPHL describing the "g" and the "h" as slender, the H isn't slender. It isn't anything but a bad attempt to replace the ponc séimithe (the dot over the g). How I would describe it is: The "gh" is slender because the "i" is a slender vowel. And I would say h is leniting the g.
I don't disagree with any of this, but I removed all the pronunciation based explanation from my answer because a) it wasn't obviously relevant to a question about téigh, and b) I was answering a question from EoinGLEJohnENG, who is clearly in too much of a hurry to deal with complex concepts :-)
I should have stuck with my original answer, which was that consonants aren't intrinsically either broad or slender, they take that characteristic from the closest vowel. Word such as verbs also aren't broad or slender, but they can have a broad or slender ending, depending on the vowel closest to the end.