I was looking at The Complete Songs of Robbie Burns and got to thinking if he actually wrote those songs or just poems and then somewhere down the road people started turning those poems into songs.
If he just wrote the poems, were any of his poems made into song in his lifetime?
I'm afraid you might have it the wrong way around - one of the things Robert Burns was doing was taking traditional songs and putting lyrics on paper, recording (and, sometimes, re-writing or just making up) what had been an oral or musical tradition.
Smithsonian Folkways covers some of his process in the liner notes to Songs of Robert Burns, but it's part of the reason why he's so celebrated in Scotland today - he was an ardent preservationist of Scottish culture, folk tales, language and music included.
In the history of poetry, this is a fairly common thing - China's Classic of Odes (Shi Jing) contains many poems that were written as words to pre-existing folk melodies, only more than 1,500 years before Burns was born. And, in the Bible, Psalms has vexed readers with references to a "lost tradition" of music.
Anyway, to answer your last question, Burns also did collaborate with contemporary composers, and some readers have argued that you can't really get some of the poems without hearing the music. Catarina Ericson-Roos wrote quite comprehensively about this in The Songs of Robert Burns: A Study of the Unity of Poetry and Music, one of the books from Jean Redpath's library. (Redpath, if you don't know, is mostly known as a Celtic folk singer... but was also pretty seriously into Burns on a few other levels as well.)