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As question suggests: what does き stands for in 招き猫? Is it some sort of particle, that "connects" 招(invitation) and 猫(cat) or something else?
No, き here is a part of the 招き - invitation. It's an okurigana, you can say that it's a reading hint to know that 招 should be read using a kun reading - まねき and not an on reading - しょう.
猫額 "cat's forehead", used as a metaphor for "small thing"