MacBook Pro - apperently broken. Working, but very slow.

I bought Macbook Pro 13 v.A1278 apparently broken. Not starting, CD stuck inside, I just saw bootcamp / windows trying to run in save mode.

When managed to eject CD windows started. I did not see mac partition, than I cleaned up the disc and installed Mountain Lion, but it was extremely slow.

I replaced RAM (4GB) to check if it's a problem, it did not help.

Cleaned the disc again and installed Lion from USB. It's running 3 times faster but still very very slow (like windows XP after 3 years of using)

I run disc utility, verified/repaired disc it was looking ok.

Battery seems to be untouched but it's not charging, iStatspro showing battery health nan% cycles 0, charge nan%, charge bar showing 80% charged.

What can be broken? if any..

How can I check what is the problem

Is there any diagnostic software I could install and run?

Any other steps I should undertake?

In Mac service, they are useless they said motherboard is broken. I do not believe that. (When I gave them my other old Mac with faulty/broken graphic card they said the screen - which works ok - should be replaced, so Mac service is not an option)