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To charge any battery the voltage of the input must be more than the battery's output.
1 watt will do the job.
I did not locate a solar panel laptop itself, but there are solar panel laptop charger for the laptop battery. these will last many years and substitute for buying a laptop battery every year.
None, since one does not normally "power" a solar panel. However a single 7w lamp would be useable with a 5watt panel but with reduced efficiency unless the panel was used to charge a battery first and then the bulb was powered by the battery.
You need to provide many more details. So assuming that the panel voltage is reasonable, the panel is clean, it is sunny and the panel is pointed at the sun. Then it will take about 225/7 hours to charge the battery. About 300 hrs. But in fact charging is only about 85% efficient (charge at 14 V get energy out at 12V or so). This ups the charge time to maybe 350 hrs or so. Charging can be improved by inserting a MPPT charger between the battery and the panel. This bit of electronics decouples the battery voltage and the panel voltage allowing the panel to be operated at its' most efficient voltage. That might knock off 15% of charge time bringing you back to 300 sunny hours.
You can find 12 and 6 volt solar battery chargers, travel solar battery chargers that fit in your backpack, automotive solar battery chargers, marine solar panels, RV solar panels, PDA and personal media solar battery chargers and even solar rechargeable batteries. Silicon Solar, Inc (http://www.siliconsolar.com) has many of them available for sale on their website. You can also find some solar battery chargers at Sundance Solar (http://store.sundancesolar.com) You can buy solar battery chargers to charge your car, to charge mobile devices like cell phones, laptops and pda's, and to charge rechargeable batteries. You can also purchase 12v and 6v battery chargers that are good for taking on camping and hiking trips, battery chargers that fit in or are part of your backpack, fixed panel chargers and flexible chargers that fold up. You can buy them at http://www.siliconsolar.com/solar-battery-chargers.html, http://store.sundancesolar.com/solbatchar.html, http://www.earthtechproducts.com/flexible-solar-pa
A solar panel is a large investment. It is also somewhat complicated to install. It would be best to have a professional install the panel. Many companies will install the panel at no charge.
A solar panel can be installed at the top of the light, pointing in the best direction to the sun. This is attached to a battery that will charge during the day and then power the light at night. This technology is already in use in many places round the world.
There is too much information there. Charging a 100 amp-hour battery fully would take 18 hours at 5.5 amps, or 6 hours at 16.67 amps. At 5.5 amps the power would be 12x5.5 or 66 watts, and this is the rating of the solar panel required. That would be about 0.4 of a square metre.
A photoelectric cell, in a solar panel. Nowadays, many calculators have a miniature solar panel, to let the batteries last longer (battery power is only used when it is too dark).
Read the manufacturer's instructions. The unit is used to charge other devices using the sun. The unit itself doesn't hold a charge. How long it takes to charge other devices depends on what you're charging. This article offers a comprehensive overview and explains how to calculate the time various devices take to charge with a solar-power charger http://www.12voltsolarpanels.net/12-volt-solar-powered-battery-chargers A solar battery charger does not take a charge, hold it and then spend it to charge other devices. The solar battery charger is connected to the device and uses solar energy to re-charge the device, as opposed to charging its own self first. How long it takes the charger to charge another device depends on the voltage of the device.
It would depend upon how many ampere-hours the battery bank can store and how often the 50 watt panel charges at full power. Obviously, if it takes 3 times as long to charge as it does to discharge (50/12 vs 1500/120), your battery is going to go flat in no time.