If captain's screens are out, and let's say that the airplane is about to land and the captain is supposed to land it, will he/she switch the seats with the co-pilot? or does he/she depend on the small little analog gauges on his side?
If the plane is on final approach and the pilot flying loses his displays the co-pilot (who is also rated to fly the aircraft) may take over. Perhaps more importantly the system is built to avoid this. The "screens" you mention are independent and both (2 screens per side) are capable of displaying the primary instrumentation/information, a situation where both fail simultaneously and completely is rare.
All screens in the cockpit are generally visible from both seats, any pilot who was a former flight instructor (and many likely were) has plenty of experience viewing instrumentation from the opposing seat. Flying and landing like this should be of little issue.
In good weather to land an airplane the most important instrument is the airspeed indicator (and perhaps angle of attack indicator if you have one). With that in mind the pilot flying may continue the approach if the backup instruments are still working and in view, that is after all why they are there.
Some airline operational specs may have specific procedures for this as well but as far as I know the FAA does not really have anything to say on the matter.
No, the captain will simply have the first officer land. The co-pilot normally lands half the flights anyway, one more isn't an issue.
If the airport is restricted to captain only landings, dictated by airline policy, they should divert to a non-restricted airport.
Pilots should not switch sides in flight because the sight picture is slightly different on each side. Additionally, adjusting to the mirrored display layout and opposite hand controls is not something you want to do in an emergency. Training captains receive special practice on "right seat" flying, but not ordinary pilots.
While the standby instruments are adequate for landing, there is no reason to give up the first officer's fully functioning displays.