You're close Kevin. Hydrogen pretty much exist in nature only as H2. since you're adding Lithium's electron to a PROTON (H+), it will only become H, which is still missing an electron from it's outer valence, which is why it goes and covalently bonds with another H, to make H2. notice the break down of H2O ---> H+ + OH-. THe oxygen has such a high electronegativity that when one of the H's leaves, the Oxygen will keep that electron, making OH-, and the leftover hydrogen atom H+. The OH- is what attracts Li+ to it to make LiOH.
This is a pretty simple explanation because of you not knowing about oxidation and reduction yet. Also if I have made any errors, anybody can feel free to let me know
Also to clarify some of your other questions - hydrogen has a higher ability to gain electrons (reduce) then lithium, so yes the H2O does get ripped apart. The products (H2 and LiOH) of the reaction would be more stable since that is why spontaneous chemical reactions occur.