Very poor installation
There is obviously a leak in either the firewall/cowl area or the window seal.
Leaky weatherstripping around the doors.
Get inside with a flash light and a friend outside with a garden hose and find out where the leak is.
Yes if it rains it will leak into the attic or overhead space until it is saturated then it will leak thru the ceiling.
vacuum leak?
One reason your floorboard is wet after it rains is that there may be rust holes in the floor or the floor pans are rusted out. A leak in the windshield seal may also be the cause of wet floorboards. Bad door seals or window seals can allow water to soak the floorboards when it rains.
well...some do but some dont....my momes does
You have a leak. Possibly the drain lines are stopped up or the leak is at the windshield.
you need to take the air vents off and when you reach up inside them you will feel leaves and maple keys or what ever is falling from your trees packed in there.It is blocking the drainage out the side of the body.
Leaking heater core
Vacuum leak
the water most likely is condensate from a clogged condensate drain if you had a secondary drain, you would see water drip from the secondary likely you have no secondary, and you are getting water out the vents if you have some of the a/c in the attic you better get the clog fixed right away, or water may seep through the ceilings note the more humid it is, the more condensation is created--hence, the correlation of your problem with rain