Roger wrote it, yes.
It apparently started out as a Queen demo during the AKOM sessions, then scrapped.
Later, it was recorded with Roger's band, The Cross, and made into two versions. The U.K. version of "Shove It" (The Cross' first album) features Freddie on lead vocals, and this is the version where most of his vocal part comes from in the later Queen version featured on "Made In Heaven" (and also released as an edited single version).
The U.S. album version, which is also the U.K. single version, is remixed and has Roger on lead vocals (leaving Freddie's backing vocals in near the end as a result).
The U.K. single has the Freddie version on the b-side.
Hopefully this answers your question on Heaven For Everyone.
And also, Living On My Own was a Freddie Mercury track, that yes indeed he did write, that originally was recorded for Freddie's "Mr. Bad Guy" solo album in 1985.
In 1993, after Freddie's death, the track was remixed and released to dance clubs, where it received enough great response to make it into a single release, which gave Freddie his only #1 solo single, also his 2nd and last post-humous #1 single whether with Queen or as a solo artist (the first one was the charity re-release of "Bohemian Rhapsody" in December 1991, doubled with "These Are The Days Of Our Lives").
I'm assuming you're probably thinking that this was a Queen track because it appeared on Greatest Hits III. It's not, as other solo tracks by Freddie and Brian appear on that compilation as well, partially hence the + in "Queen+".