1) Barristan Selmy stated in AGoT, that he fought beside Prince Lewyn of Dorne. How was Lewyn related to Doran Martell and was he the seventh Kingsguard who was replaced by Jaime at Harrenhal?
Lewyn was an uncle to Doran Martell. He was a Kingsguard, and died fighting on the Trident.
2) What would happen if a Kingsguard is seriously crippled, thus that he is unable to protect the king anymore?
With seven Kingsguard, there are usually enough who are young and strong to allow older members to serve our their lives with honor.
3) Kingsguards may not have a wife or children, but does this also include a vow of chastity?
In theory, yes, but at least one Kingsguard was executed for sleeping with a king's mistress, and many others have doubtless had "lapses."
4) Does the oath of a Kingsguard include to serve _whoever_ is the king, even if the new king rebelled against the old one, or did Jaime and Barristan _choose_ to continue their service as Robert was crowned?
The oaths did not envision rebellion, actually. Robert pardoned Barristan and Jaime, and they accepted the pardon and continued to serve.
5) Why were men like Meryn Trant, Boros Blount, Preston Greenfield and Arys Oakheart ever accepted as White Swords? Nobody thinks much of their skill.
Sometimes the best knights are not eager to take such stringent vows, and you have to settle for who you can get. Other factors also enter into the choices -- politics, favoritism, horse trading, rewards for past service, etc. It's a plum appointment for a younger son, or a knight from a minor house. Less so for the Great Houses. Also, Robert had five vacancies to fill all at once, an unusual situation -- imagine the nominations we might get if six of the nine members of the Supreme Court all died within a few months.
Institutions like the Kingsguard change over time. The original Knights of the Garter were warriors all, the strongest, bravest, deadliest men of their time, with an average age under thirty. The present Knights of the Garter are octagenarians, and their parades are processions of wheelchairs and walkers.