Dissolution of Lithium in Methanol
I have not done it with lithium but many times with sodium. Lithium is the less reactive, so it should not give more problems. We use sodium to dry methanol, so it is a common operation in the lab. The reaction is:

$\ce{2Li + 2CH3OH -&gt; 2LiOCH3 + H2}$ 

Of course, hydrogen is dangerous, especially in the presence of oxygen and still more in this exothermic reaction. So, be careful. 

The solution is stable, but attracts water and carbon dioxide, so it must be stored well closed.

To get dry lithium methoxide, the easiest is just to evaporate the excess of methanol.
You put it in a round bottomed flask, attach it to the rotary evaporator, open the vacuum and distill it of. With sodium it works perfectly well. A white solid remains. Again, this solid will attract water and carbon dioxide, so it must be stored properly. 

Finally, if the only purpose is to have a methanolic solution of lithium, you can better use lithium chloride which is very soluble in methanol, and much safer (no reaction, no hydrogen, no heating).