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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". ...
The best way is to look each of these terms up individually in Japanese language dictionaries and check examples of usage, but here's a translated synopsis. Many of these meanings overlap. でも: "though that may be the case" / (though the prior statement may be true) しかし: "in contrast to the previous statement" / (lit. "unlike" the prior statement) ただし: is ...
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