In reply to:

I'm trying to remember what the Westron word for Shire is

I know this - it's Sûza.

Tolkien maintained the conceit that he was the translator and not the author of LOTR and The Hobbit, and that his decision to represent Westron as English (and the language of Rohan as Anglo-Saxon) was so that the readers' experience of language would match that of the Hobbits. He therefore anglicised the personal names and place names of the Shire, with the exception of certain names from Buckland such as Meriadoc, which have a Welsh flavour to represent their Stoorish origin. He did, however, provide several examples in the Appendices to LOTR of the "original" Westron versions.

There is no doubt that the phonology of Quenya is very similar to Finnish. I don't believe there is any lexical commonality, however, and I have no idea whether the grammatical structure is similar.