yes. but more likely you just need a tune up. though it could be a lot worse too.
It sure can. In some cases your car will not run at all. Normally bad gas just means it is contaminated with moisture. A tiny amount of moisture and your engine will miss and run rough.
Because when you go uphill or put "load on the car" this is when the car will miss under load & "not miss" when you ease up on the gas as you are not stressing the engine. As to why this happens can be many thibgs - spark plugs, ignition coil bad or bad plug wires & many more to many to list here
It can.
It will make your car use more gas than it should
Yes Yes
possibly yes, with a bad o2 sensor you can have the wrong gas/air mixture making it harder for the sparkplugs to ignite the gas.
Filling your fuel tank will not make your car shake, unless you got some bad fuel that was contaminated with moisture.
There are several things that could be going on. The most common thing that would make your car smell like gas when the heater is on would be a bad heater core.
Yes, it is possible as you may have a miss. That miss would turn on the check engine light.
Yes, it sure can.
Battery? / Alternator? It's bad gas?
It starts running bad and changing the fuel fixes it.