Roarers, also known as Recurrent Laryngeal Neuropathy or Laryngeal Hemiplegia, results from damage or neuropathy to one of the recurrent laryngeal nerves (usually the left hand side). As a result of the nerve damage, the muscle (the dorsal cricoarytenoid muscle) that moves the arytenoid cartilage (see the pics) ‘in’ and ‘out’ is affected and begins to atrophy. This hinders the ability of the arytenoid cartilage to move.
To answer your question about whether the horse will be fine for what you want to do with him (am I right in assuming this was your question?), firstly there are 3 different grades of roarers (grade 2 – 4, with grade 1 being a normal horse).
Usually, with non-racehorses, the only complaint is the sound which is caused by air turbulence as it passes over the affected vocal cord and the tie back procedure would have corrected this (it is used to increase the airway space and exercise capacity). It is usually only in racing horses which exercise to maximum capacity that exercise intolerance is noted. I highly doubt that him being a roarer will affect what you wish to do with him, especially as he has had the tie-back procedure already. The cough may, or may not be related to him being a roarer.
The tie-back procedure is also called a laryngoplasty and basically, all that they do is permanently tie the affected arytenoid cartilage back so that it is always in the ‘fully open’ position (see the pics)
I have attached two pictures which hopefully help explain it. If you need any more information, I shall try my best to help!