I'll try to explain this by giving some examples. I'll assume that the person staying over / letting someone stay over is the speaker (unless otherwise specified) to make this easier to read.
I assume that by「~てしたり」you mean「~たりした」, i.e. the past tense of「~たりする」. As you say, the「~たりする」corresponds to something like 'and stuff like that'.
In this context「泊める」means "to have/let someone stay (at one's house etc)" while similarly,「泊まる」means "to stay (at someone's house, a hotel etc)". The basic forms for these with「~くれたりした」and「~もらったりした」are:
Normal interpretation: "A friend let me stay over and stuff" or more lit. "I had a friend let me stay over at their place and stuff".
Normal interpretation: "A friend let me stay over and stuff".
Normal interpretation: "I had a friend stay over at my place and stuff".
Normal interpretation: "A friend stayed over at my place and stuff".
As for the other forms that you mentioned, I assume you meant (some of) the following (without the「~たりした」/ "and stuff like that"):
Normal interpretation: "I had a friend let me stay over at their house".
Alternative: "I stayed at a friend's house (without their permission)".
Normal interpretation (although weird): "I let (someone) stay at a friend's house (either with the friend's permission or without, depending on context)".
Normal interpretation e.g.: "I made ("forced") a friend to let me stay at their house"
Alternative: "I made a friend let (someone) stay at that friend's house" etc.
Normal interpretation: e.g. "I made (someone) stay at my friend's house".