Is it normal for rain to be leaking from the chimney heating vent when it rains?
Answers:
It shouldn't be leaking, call whoever installed it and tell them to come back and fix it. They should do it at no charge if they are reputable.
Other answers:
not normal
not normal
No, of course not, and the installer owes you a repair. Having said that, your question is confusing. There is a chimney, liner etc. (wich is not a "vent") and there is the distribution system for the room or heated air and there are vents in that system. Your question confuses the two.
Check
a) The brickwork of your chimney. Even small imperfections / cracks in the pointing can lead to leaks
b) The flashing around your chimney; often not done properly.