Value of a car for total loss

The value of a totalled car is its then-current (that is, at the moment it was destroyed) fair market value, often referred to as its "blue book" value, based on its make, model, age, mileage, options, and condition. The purchase price has nothing to do with this value: a 2016 Sentra S, for example, has the same value whether the buyer negotiated a poor deal and overpaid, a great deal and underpaid, or was even given the car as a gift. Consummer reports, the Kelly Blue Book, CarFAx...there are a number of different services or products which can provide estimated values for cars, and if you feel the value offered is too low, you can certainly provide other evidence, if you find other sources supporting a higher value, that you should be offered more and try to negotiate a higher price. Ultimately, though, if they refuse to offer what you think was the fair, then-current market value, you'd have to sue for the money, which may or may not be worth it, depending on how much that difference is.
It is common, by the way, for cars to *sharply* depreciate or drop in value as soon as they are purchased; the assumption is that as soon as a car is not in mint dealer condition, it starts to accumulate wear and tear and is worth less. 
The other side's insurer is not required to voluntarily offer to pay your rental costs; they can decline to offer rental reimbursement if they feel, for example, that you delayed taking a reasonable offer or settlement. You could try to sue for this money, too, but it would probably not be worth suing for the rental costs alone, unless  you were also suing for other money (e.g. the difference in value) as well. If you were to sue and win, you could likely recover rental costs for a "reasonable" time post-accident--the time it would take a reasonable person to acquire a new car.

The value of a totalled car is its then-current (that is, at the moment it was destroyed) fair market value, often referred to as its "blue book" value, based on its make, model, age, mileage, options, and condition. The purchase price has nothing to do with this value: a 2016 Sentra S, for example, has the same value whether the buyer negotiated a poor deal and overpaid, a great deal and underpaid, or was even given the car as a gift. Consummer reports, the Kelly Blue Book, CarFAx...there are a number of different services or products which can provide estimated values for cars, and if you feel the value offered is too low, you can certainly provide other evidence, if you find other sources supporting a higher value, that you should be offered more and try to negotiate a higher price. Ultimately, though, if they refuse to offer what you think was the fair, then-current market value, you'd have to sue for the money, which may or may not be worth it, depending on how much that difference is.
It is common, by the way, for cars to *sharply* depreciate or drop in value as soon as they are purchased; the assumption is that as soon as a car is not in mint dealer condition, it starts to accumulate wear and tear and is worth less. 
The other side's insurer is not required to voluntarily offer to pay your rental costs; they can decline to offer rental reimbursement if they feel, for example, that you delayed taking a reasonable offer or settlement. You could try to sue for this money, too, but it would probably not be worth suing for the rental costs alone, unless  you were also suing for other money (e.g. the difference in value) as well. If you were to sue and win, you could likely recover rental costs for a "reasonable" time post-accident--the time it would take a reasonable person to acquire a new car.