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Only the front turn signals act as running lights, the rear turn signal lights are not running lights and should not come on unless turn signal switch is activated. As for the brake light, I assume it doesn't work with either front or rear brake applied. That being the case and if you are certain the fuse is good then the most likely culprit is a bad ground in that circuit. The most likely place to look for that is at the bulb itself. Often times the socket that the bulb sets down in becomes corroded and or dirty. Take the bulb out, clean the contact points on both the socket and the bulb and try it again. The relay should have no effect on brake light working.
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