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As you said, 行ってて is the colloquial contraction of 行っていて and we almost always use the contracted form in informal speech. There is a fairly big difference in meaning between 先に行って and 先に行ってて. 先に行って simply means "You go first." 先に行ってて means "You go first and wait for me/us." or "You go first and I/we will follow you."
It is exactly as you suspect. It is totally illogical to consider 固い・堅い・硬い, for example, to be three different words and here is why. Whenever you are dealing with a kun-reading word, you need to remind yourself that it existed when Japanese was merely a spoken language. We had no way of writing 固い・堅い・硬い or even かたい. All we had was the sounds "katai". ...
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