When seeing なんだ, the first question I ask myself is, “Is this な + の + だ?”
霊感少女なの = acts as an adjective nominalized into a noun, to refer (in this case) to a person. “a girl with a sixth sense”.
The core of the sentence, with (as context tells us) Rei as the subject, is “Rei is a girl with a sixth sense” (to use the English term used for someone who can see spirits).
I’m don’t have enough knowledge yet to say why it does stated this way, using の.
For the whole sentence (which I’m assuming each word balloon is a separate person speaking), “But, she’s a girl with a sixth sense, right?” or a little less literal with the grammar, “But she has a sixth sense, right?” The use of でも does confuse me, because でも is used to contrast with what was said before it, and this sounds like it’s following the conversation.
Perhaps something like: “She a shrine maiden, after all.” “It should be understood by such a person, the children’s location.” “But, she has a sixth sense, right?”
I do wish I had my official English (second translation) on hand to compare with, but I won’t have that out of storage for two more weeks.
This なんて I read as “(things) like”, as you’ll find in some dictionary. “Things like a sixth sense…” or “Something like a sixth sense…” which leads into her saying it’s not really for finding things.
せっかく and のに are used together when great effort has been put into something, yet that effort may be going to waste. The effort appears between the せっかく and のに, so here the effort they have done that has been wasted was 「この神社に来てやってん」. Coming to the shrine was a waste of effort. “Even though I went to the trouble of coming to this shrine…”
This し is a conjunction particle, separating sentences like “and” in English. The と makes 意外 into an adverb, so it applies to the following verb (やる). I believe 意外と can be used to mean “strange” or “weird”. So, (loosely) “I haven’t seen your grandpa around, and you all do those weird prayers in your family.”
I am still a learner, so I may have made mistakes.