Puedas jugar?
It means, "the chess." Frequently, Spanish uses definite articles in ordinary speech which are not translated into English. Example: ¿Te gusta jugar al [a+el] ajedrez? Do you like to play chess?
Playing tennis in Spanish is jugar al tenis.
"ajedrez" is "chess" in English.
There is no difference between: Jugar al beisból and jugar el beisból. It depends on the country, but both forms are used.
This sentence means "Do you prefer playing baseball or soccer?" It is asking which sport you would rather play.
To play tennis.
"Can I play basketball?"
jugar al baloncesto
To play the white. Possibly in chess?
As a statement "you like to play tennis", it would be "Le gusta jugar al tenis." As a question "do you like to play tennis?", ¿Le gusta jugar al tenis?" Spoken, the differrence is just in inflection - with the voice rising at the end. If you are asking someone if they want to play tennis, the it is "¿Le gustaría (Or "Quisiera") jugar al tenis conmigo?"
"I love to play basketball" in Spanish is "Yo amo jugar baloncesto". It is pronounced "Yo AH-mo hoo-GAR bah-lone-SESS-toe". Please see this site for confirmation of the translation: http://www.answers.com/library/Translations