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A: どこですか？住んでるとこ
B: 私地元は逗子っていって鎌倉の方です。
Firstly, is there supposed to be a ろ after the end of 住んでるとこ(as in, ところ, and was it cut short due to conversation?)
Secondly, does 言う in the te form simply loosely connect the first clause with the following clause (A place called Zushi, near Kamakura) - like it doesn't mean 'and' here, it's more like a pause because she still has more to say?
Firstly, is there supposed to be a ろ after the end of 住んでるとこ(as in, ところ, and was it cut short due to conversation?)
It was indeed shortened from ところ, but とこ is an extremely common abbreviation of ところ used in colloquial speech, so I wouldn't say that it was "supposed to be" ところ. This is mentioned briefly here (for an entirely different usage of ところ), but it's also listed as an alternative form of ところ in Jisho. If you're comfortable reading Japanese, you can also see here.
Secondly, does 言う in the te form simply loosely connect the first clause with the following clause...?
The verb 言う here is in the て form so that the first clause can be connected to the second, but the verb itself has semantic content here - it's not just a connector. This 言う is acting like "called" does in English when it's used to explain the name of something. See here and also Jisho.
私（の）地元は逗子っていって鎌倉の方です。
My hometown is called Zushi, in/near Kamakura.
