Well, the first two questions are fairly straightforward. You can buy a device to test your batteries here, and specific gravity is explained by Wikipedia.
As for the golf cart, though, this is a rather subjective question. There are pros and cons.
Reasons to keep your golf cart charged
You may be invited to play golf with a personal hero at short notice. Imagine your disappointment at having to turn down a last-minute invitation to spend the day with Black Beauty? The opportunity of a lifetime may pass you by.
What if you need to go somewhere, and your legs suddenly don’t work?
Golf carts are useful in case of zombie apocalypse. They’re not very sturdy but will just about hit a top speed faster than a zombie, and will carry you further than running. It’s also easier to shoot while driving than on the hoof.
If the power goes out, you can use the golf cart battery’s power for a short time.
You may simply want to play some unplanned golf.
Reasons to keep your golf cart flat
You may be invited to play golf with a personal enemy, or some other shady character best avoided. Should Bad Horse, the thoroughbred of sin, call you up for a golf day you will need a good excuse to pass on his invitation.
Should electricity-sapping aliens invade earth, they will pass you by.
If the poor lumpen-proletariat stage a coup, they will look to depose those who have rich man’s hobbies, such as golf. You can fool them by saying you won it in a raffle and never used it.
You will be certain that nobody can embarrass you by running you down with your own golf cart.
If someone asks to borrow it and you don’t trust them, you can legitimately turn them down without sounding selfish.
As you can see, the decision is all yours. You just have to weigh up your options.