If you are going to jump start someone else's vehicle your car should be running. Having said that I will second namantm that you should get a jump start battery pack. One of the newer lithium ion type that you can get from $50 to $100. They are really great. You do not have to worry about lining up the cars. You don't have to worry about potentially damaging your car. They are smaller than a set of jumper cables. For yourself, you don't have to rely on another car for the jump or to get your car in a place it can be jumped. They have other uses like integrated flashlight, power bank for recharging phone, etc.
The first time I used one was great. I saw a family with a dead battery in a minivan at a kids soccer game as I was walking to my car. They were calling AAA. Normally, I would not offer to jump becuase their car was in a horrible place to line up and I don't know if I'm going to get into a struggle with the dad wanting to do the jumping and I don't let people touch my car. But, I had the jump start so I told them I had a jump start and I could start it if they wanted. Took like 20 seconds to setup and start the car. They were totally happy. I started giving these things out as gifts to family and friends.
Lithium batteries do not like heat so store this in the coolest part of your car, low in the trunk or under a seat. They may not work great in extreem cold either but you can warm them up by putting in your jacket for a few minutes. You should check it every few months and make sure it is near full. 80% full will give you the best battery life in the long run and you can get plenty of jumps out of that.
I live in Miami and I get 2-3 years out of a car battery. Heat kills these things and they just bake in the engine compartment. Same battery brand in the Miata that has the battery in the trunk lasts 7 years. When they start to go it will seem to work fine but something as simple as leaving the hazards on for 20 min while you have the car in the street while you mow the lawn can lead to a no start situation. So, using the car to jump another vehicle with your car not running could definitely leave you in a no start situation with a weak battery.
Rearding the spinning sound. Normally when the battery is near dead and you cannot start you may here a series of clicking noises but the car will not run the starter. When you start the car a solenoid pulls the starter motor gear into the flywheel to engage the engine. If the solenoid does not actuate you may here the motor just free spinning without the engine turning. That sounds more like your starter motor solenoid is failing or the electrical connection is poor. If the car starts with a jump starter or jump from another vehicle its the battery. If it does the same thing you have an issue with the starter. The solenoid can usually be replaced which is much cheaper or you can replace the whole unit starter and solenoid. I would check the fuses and electrical connections first to make sure that is not the problem before getting into replacing the starter solenoid.