A Home For Su-lan? (first published in 1971)

And Man Without a Face was first published in 1972. But I really think Sticks and Stones is your book.

Sticks and Stones was published in 1972

or possibly
Sticks and Stones -- actually, I think this is probably the one.

Your description makes me think of
The Glass Sentence but it was published in 2014.
Girl at Sea - link to Ket's suggestion, which is a solid one.

Also, putting this up here as an example of possible illustration style.
Leo & Diane Dillon illustrated a ton of books of african folktales. If this style of art looks familiar it might be another anthology they worked on:
Behind the Back of the Mountain: Black Folktales from Southern Africa

Librarian here - this is a cataloging issue. The bibliographic record should have a 246 tag, indicating an alternate title, so that the book shows up in the catalog regardless of whether you search for the old or new title. Unfortunately, many libraries have reduced or cut their cataloging staff, and rely on books that come pre-cataloged by the book vendor, leading to problems like this. (The more you know!)

I think this may be
Haphazard House - the father buys the house with money he wins from betting on horses, and there's a ghostly figure at the window.

I know you said it's not time travel, but I'm going to throw this one out there just because the travel happens through an "egg shaped device" and some of the editions have green covers:
The Green Futures of Tycho

This is a bit of a long shot but
Away deals with Irish heritage and folklore and has a bit of magical realism but I don't think there are mermaids.

Possibly the Silver Lake trilogy? The first is
Summer Love

This is possibly
Saturday, the Twelfth of October. I don't think she falls from a tree, but she falls asleep in Central Park and wakes up in the very distant past.

Mary beat me to it by 5 minutes. I second Coldwater.

Possibly
The Survivor? Also by Robb White. One of the GR reviews is by someone who got it from a Scholastic Book Fair...