Me covers me, to me, for me, from me and myself. The same goes for te and nos. Spanish tends not to need these words as much as English does.
me habla = he speaks to me · me lo compra = he buys it for me
Do not let the to confuse you about where the word goes either. It still goes before the changed verb.
Me habla · Me hablan · Me lo hablan · Me lo compra · Te lo compro · ¿Me lo quieres hacer? / ¿Quieres hacérmelo?
Shall in English is really only turning the sentence into a question, so in Spanish you just make it sound like one.
Quedar is to remain or to be left over, and it is quite a flexible verb with several other meanings. To say I stay, Spanish says I remain myself.
The e does not split here.
Me quedo aquí hasta las diez · Nos quedamos hasta las nueve · Te quedas aquí · ¿Te quedas aquí?
Resist any temptation to read quedar as /qwedar/. The combination qu in Spanish is always a k.
At six o'clock is a las seis. What you actually mean is a las seis horas, "at the six hours", a little like the English "at sixteen hundred hours". People said the horas historically, but not any more.
The month names are the hook: in the Roman calendar there were only ten months, so October was month eight and December was month ten. Nueve and diez are both splits, from nov- and dec-.
He cocinado a las seis · Hemos cocinado a las ocho · Diez sounds /dieth/ in most of Spain.
Cocinar is simply a verb made from cocina. Hasta la vista is literally "until the view", until we see each other.
Cocino · Cocina · Cocinas · Tú cocinas · Cocinan · Ellos cocinan · Ustedes cocinan · Cocinamos · Nosotros cocinamos · ¿Me puedes cocinar algo? / ¿Puedes cocinarme algo?
In English it sounds much more natural to say I cooked at six o'clock. In Spanish there is no problem saying I have cooked at six o'clock.
He cocinado a las seis