Battery capacities are usually provided in amp-hours or milliamp-hours. A quick search shows that typical car batteries have a capacity of around 45 amp-hours. That means with a one ampere load, it has enough energy to run for 45 hours. This isn't the most accurate way to think of capacity, since the total energy available depends on the manner in which it's used (quickly? slowly?) but it should suffice for some back-of-envelope calculations.
So to estimate the runtime, divide the capacity in amp-hours by the average current requirements of your device. If your device requires 50 mA on average, then:
\$ 45Ah / 50mA = 900 h \approx \text{37 days} \$
Most car batteries are not designed to be deeply discharged as part of normal operation, so avoid that. It's hard to say if your car has any sort of protection against running the battery down enough that it won't have enough to start. It would depend not only on the car, but also on which circuit the device is connected; some may have protection while others may not. I know my car does not offer any protection (even the dome light will run the battery down), but as the car in question gets newer and more expensive it becomes more likely there will be some circuitry that will do something intelligent.
If your device has batteries of its own that will power it should the main power be lost, then you can simply add that runtime to the estimate runtime of the main battery.
It's hard to say how long it will take to recharge the battery. It will depend on the charging method (fast charger? slow charger? the car's alternator?) as well as the type of battery, but something an hour and 2 days is likely. See this page from Interstate Batteries for more information on that.
Since you're new to EE, I'm going to have to say that the answer is no. What you are asking is pretty straightforward, but it's not something you can do with hardware-store parts.
Basically, what you need is an energy storage device that you can insert between your car battery and your computer. This is complicated by the fact that you need to isolate this storage from variations in the battery voltage. That is, when you try to start the car and battery voltage drops, you don't want your storage device to try to backfeed the battery, as this will exhaust the storage device and your computer will die. In principle, you can do this one of two ways:
simulate this circuit – Schematic created using CircuitLab
Note that I'm showing your storage device as BOTH a capacitor and a battery. This is not necessary - you can use either. Using a battery instead of a capacitor is in some ways easier, but you have to worry about keeping it charged, which can be a royal pain in the whatzit. On the other hand, you don't have to worry about making sure the capacitor is charged before starting the car, which can be a problem, too.
Also note that the second approach, storing energy after the converter, seems simpler because you don't need a diode to keep from backfeeding the car battery, but this may or may not be true, depending on the converter. It assumes that the converter will not let current flow from output to input when the converter is not working. This may or may not be true. If you select a battery, it also assumes the converter will turn on properly with a humongous capacitor hanging off its output. This also may or may not be true.
Using a capacitor will work, but you have to size the capacitor properly. To do that you have to know a) the current it supplies when needed, b) how long it needs to provide the current, and c) what voltage drop is acceptable during that period. If the answers are a) i amps, b) t seconds, and c) DV volts, then
C = 1,000,000 x i x t / DV (in uF)
So, if you're supply 2 amps for 3 seconds, and you can accept a voltage drop of 2 volts,
C = 1,000,000 x 2 x 3 / 2
= 3,000,000 uf, or 3 F.
You'll have trouble finding a 3 F, 12 or 24 volt capacitor.
So, like I say, this may be rather more trouble than you can handle at this stage in your education.
Best Answer
Even though the jumper cables are very thick they often can't supply all the current needed by the starter.
If you wait a while for the battery in the dead car to charge even through the jumper cable from the good car it can provide some of the starting current to assist the process.