Does Google penalize if we get EDU backlinks?
I have created some Do Follow & No follow EDU backlinks to my website and I saw that it has been already indexed. Later on, I read some articles about EDU links that if numbers of EDU links are pointing to our site than Google does penalize.
No not at all, they are some great links especially if they are from local colleges in your area. They can put your website in a position to rank high
Yes, Google does absolutely penalise sites for incoming links, and the practice you allured to - "anyone could pay someone to lower his competitors' ranking" - is known as 'Negative SEO'.
Google penalises (or, if not strictly 'penalises', then certainly doesn't help) sites for incoming links that appear spammy, unnatural or paid for.
Bing recently released a "Disavow Links Tool" on Bing Webmaster Tools that allows webmasters to say 'I wish to distance myself from this link' and then Bing will ignore it.
Google currently does not have something similar. But, not all bad links will hurt you; Google does already simply ignore 'bad' links. However, we are still waiting for their version of Bings Disavow Tool.
If you need advice on what to do if you think you have a bad link profile, I'm more than happy to help :)