- Does that mean that I'm limited to 800 lumens?
- Could I replace the 60-watt incandescent with an 6-8-watt and have the same brightness?
- Could I replace the 60-watt incandescent with a 25-28-watt LED and have about 3 times the brightness?
- Can a lamp take any bulb type? Why would a lamp specify that it takes incandescent?
1) no, 2) yes, 3) almost certainly yes, 4) most of the time yes but see below.
The 60W limit on the lamp is to prevent it overheating and catching fire. That means that you could, in principle, put a 60W LED in the lamp and it would be OK, or at least the lamp shroud would not catch fire.
An LED must run at a much lower temperature (about 25-50C) than an incandescent (2200-3300C at the filament!), which means it must have much better airflow available to it.
A lamp housing with poor ventilation would probably have no significant bad effect on an incandescent bulb, but it might cause an LED or CFL bulb to overheat and have a very short service life. So that's one possible reason they would specify incandescent, and note that the lamp you've linked has no way for hot air to escape the top. Also if there is a dimmer in the lamp or if the lamp contains an SCR-based (solid state) power switch, that would be incompatible with most LED and CFL bulbs.
In the case of that amazon listing though, I think it's just telling you that it comes with an incandescent bulb, not that it is incompatible with LEDs.
1) No, you can put as many lumens in there as you want. The fixture's electrical and heat handling can only deal with 60W of electricity though.
2) Based on the table you provided, yes. But in general, just get one that is 800 lumens. In the future, the actual wattage will be even lower!
3) As far as the fixture cares, you can put up to 60W of any source in there, but there are some practical limitations of LEDs that I'll talk about below.
4) It's strongly implied that the fixture is a Medium Edison Screw or E26 type fixture. This may be to differentiate it from fluorescent-only form factors that take the circular tubes.
Power, heat, and airflow: There are limitations to replacing incandescents with LED or CFL lights. Many fixtures that were designed for incandescent bulbs are completely closed and do not allow enough air to circulate. This causes the lifetime of CFLs and LEDs to be significantly reduced (not 10 years!) Unless the LED bulb is rated for an enclosed fixture (they do exist) it's not going to work well.
Of course you probably know about color temperature, that is generally available from 2700K to 6000K. The lower the number the more yellow the light (like an incandescent) and the higher the bluer the light (like older fluorescents, the moon, or the daylight sky). Most of the time it's nicer to have 2700 to 3000 for evening lighting and 6000 for daylight lighting. Keep in mind that when you look at them in the store your perception is going to be strongly influenced by what color the light in the Home Depot is. You might want to try a few in your home before committing to a big buy.
Another thing to look out for is Color Rendering Index or CRI. This is a figure that tells you how accurate colors will look under the light. It may not seem important, but food can actually look pretty bad under low-CRI light. In general, shoot for more than 80 CRI for general lighting and more than 90 CRI for kitchen and dining rooms. You can get bulbs that are 70 CRI that are cheaper, but they really are only suitable for commercial lighting like warehouses.