Looks like battery, doubt is alternator problem. Alternator will throw a warning light. Instead of removing battery and send it to get tested rather buy a voltmeter and test battery and alternator yourself, its easy to: How to test battery and alternator.
If battery is dead, charging wont help it.
Check Obd2 codes to be on safe side but if check engine light is not on, then it should be fine. There 100 things that can cause a check engine light and most of those does not make the car undrivable, you can still drive with check engine light on, however it has to be repaired as soon as you can. Check engine light.
You got to charge it using a battery charger or leave the car running, it will take few hours of idling to recharge battery. Get a voltmeter and test it after recharging (test on crank too), maybe battery is no good and cant be saved. How to test battery.
Which sensor? If you have have a check engine light on then you got a problem and 99% of times is never about the fuel. Got to fix it as it might cause bigger problems down the road, if you have a problem with Oxygen sensor for example, driving it like that will eventually destroy catalytic converter.
You never know with batteries, but if they tell you to replace I would assume they have a reason. However, you can test it on crank yourself and see how low voltage drops, anything under 10 volts is replace soon, close to 9 volts or lower is replace now. How to test battery.
Very likely a dead battery. Buy a cheap voltmeter and test it, its quite easy: How to test battery.