Legally your landlord is in violation of your local fire code. If they do not repair the situation tell them you will be writing letters to your local renter's association, DA, your local building board, your local firefighters, etc. The problem MUST be repaired, or the building will not be considered habitable.
Most places have rental housing services or similar:
http://www.a2gov.org/government/communityservices/planninganddevelopment/housing/Pages/default.aspx
Every rental unit requires a permit, and they perform inspections on demand.
EVERY state/county/city/township/bourough/etc has rules regarding fire safety that include proper wiring of receptacles.
1) Damage
Yes, you are more susceptible to lightning/surge problems with ungrounded plugs. There really isn't much you can do about it, other than using a long extension cord from the one grounded outlet that covers all your expensive/critical electrical devices. You can't create a "virtual" ground, so you have to make your landlord fix the situation.
2) Safety
This is a fire hazard. Many electrical devices depend on a good ground in order to avoid fires. Without a good ground, the equipment may not fail faster, but when it does fail it may do more damage as the current might flow unrestricted to other items in the house, rather than to the case and to ground.
Worse yet, though, is that you are placing yourself in great danger of electrocution. Most equipment relies on the ground to make sure the case, screws, and any touch-able metal parts are safe for human touch. When they aren't properly grounded they tend to float at tens of volts, and sometimes hundreds of volts. Touching them may shock you depending on a variety of other factors, even if the equipment is operating correctly.
However, if the equipment goes bad, you are not protected from the full line current - which would normally be shunted to ground long before you could touch it.
I recall using a bad extension cord with no ground once to power a computer across the basement - lee than 25 feet from the properly grounded outlet. The computer was electrically hot - if you touched the metal case you would feel the current pulsing through you. I eventually found the reason and cut the cord up (if you just throw it away someone might find it in the trash and try to use it).
You don't want to mess with this stuff.
A GFCI will only protect you from equipment failure which allows the current go from the line to the case, through a person or another object. It will not prevent many types of electrical fires, though it will prevent some.
An arc fault circuit interrupter would give you a little more coverage for fire issues (
http://en.wikipedia.org/wiki/Arc-fault_circuit_interrupter) but I'm not sure that you can buy those as inline plugs, they seem to be available mostly as circuit breakers for electrical service panels.
Even though, technically, your landlord can't kick you out for bringing this up, if they don't fix it and you do have to alert the authorities, then it's likely that your lease won't last long, and they'll find reasons not to refund your deposit. Go to your landlord, though, it's simply not worth the risk. You don't need to threaten them - at least not at first. Just let them know what you found out, and that you believe it's putting you at risk of fire or personal injury, and you need them to fix it. Put it in writing. Put it in writing again and send them a certified letter - which should get their attention. If they don't come out in a timely manner, remind them of the issue, and indicate that for your own safety you'll be alerting to your local rental permit authority. Then get ready to move - if they don't respond quickly to this risk, they are very likely putting your life at risk in other ways that you don't know about.
Regardless, make sure it's fixed, or that you alert your housing authority so the next renter - who may not even notice - doesn't end up in the newspaper as a crispy critter in the third page news.