Does Allah ever let your enemy become your friend, so that you may encourage each other to become better Muslims, or does your enemy remain an enemy?
How do you define an enemy? Please allow me to urge you that we should not conveniently label anybody as our enemy, be it on the basis of religion, social issues, professional relations, etc. As you very rightly imply, it will not be possible to preach, share and as a result, become better Muslims if we proclaim enmity towards each other. Everybody needs to be approached in a friendly, amicable manner; the opposite of which will only repel them. Just as we ourselves would like to seek advice when we feel we have erred, and to be given another chance; or just as we bank on others to share their knowledge with us so that we may benefit as well, others should be granted their space in a likewise manner. This holds for our dealings with both fellow Muslims as well as non-Muslims (unless they openly declare their intent to corrupt or ruin us).
By all means, one should attempt to develop trust on a mutual basis and proceed appropriately. Even when one is on undesirable terms with another due to reasons other than religion, circumstances can be made more favorable if you declare peace for Allah’s pleasure.