You have several questions here.
Is the goal to recover data from the original laptop, to repair the laptop, or charge a bare notebook battery?
If #1, pull the harddrive and install it in an empty USB enclosure.
#2 and #3 are related. Laptops require a fairly "intelligent" charger that is built onto the motherboard. Unless you have the equipment and experience for SMT soldering, that's "a job for trained professionals. Do not try this at home".
#3. It shouldn't be too hard to design and build a smart Li-Ion battery charger - if you have a fair amount of practical electronic experience.