Will compulsory voting improve the quality of a Democracy?

No, but having way more than 3 relevant parties does

Mandatory voting would put an end to at least some forms of voter suppression. That is, the state has a responsibility to get a ballot from each person eligible to vote rather than it being incumbent upon each individual to figure out how to he/she can manage to vote. 

If people don't care enough to vote there is no reason to force them.

No.  But when you get your drivers license, you should automatically be registered to vote.