1) Download Malwarebytes Anti-Malware from their website (its free) and run a scan. If it wants to clean up something let it. after the scan, try turning on your regular antivirus.
2) BSODs have codes that help debug them. If you can post a screenshot of the bsod or if you have noted down the code somewhere, let us know, it'll help.
3) Resolution depends on your graphics driver. Its possible your driver is corrupted. Try downloading from the manufacture's website or restore from a copy you might have saved.