It seems that two plumbing jacks on my roof do not have a cover on top, and I used a DJI drone to fly over, and cannot see cover on the surface too. Is it normal to have a plumber jack without cover like this? Is there any concern that some animals or rain can get through it?
Plumbing vents (in the USA, anyway) are normally not covered.
Rain will end up the same place as any other wastewater in a properly built system, and does not amount to much, given the size of the pipe.
Animal entry is not unheard of, but is rare. In the unfortunate event, they also normally end up (dead) where the wastewater goes - I've see a sewage pump jammed by one - exactly once. Not a common occurrence.
The vent freezing up in winter is one reason NOT to put a cap on it. That becomes much more likely with additonal restriction and cold surface for frost to form on.