If one is careful and keeps the 7 deoraita days of nida but not the additional zav (chumra) days, is this a problem? Is there a punishment? (I understand this is not for everyone and especially those who are confused about the counting.) Thank you.
I really wouldn't call them "chumra" days. They're rabbinically required. And they protect you in case any period is at all funny, you could wind up in a zava situation -- or nida-still-within-the-seven -- without knowing it (in which case mikva and everything accomplishes nothing, it would still be liable to Karet.) It's not just about being "careful." If you're not incredibly knowledgeable, a lot of problems can easily happen.
In theory if a person was lucky and in their situation turned out to be a simple nida start-to-finish, ending correctly at 7 days, then they would have violated the rabbinic prohibition to always wait seven clean days. The Torah commands us to follow the rabbinic enactments, though it's debated by the commentaries whether every rabbinic prohibition therefore carries this Biblical command as well, or whether it simply applies to the general framework.
Generally speaking, intentionally violating a rabbinic prohibition could have been punishable by lashes. (The Talmud talks about lashes for rabbinic prohibitions, but the commentaries have different understandings about how many lashes and in what circumstances; a lot may have been up to the rabbis judging each case. No rabbinic court today uses corporeal punishment, so the takeaway is simply: "don't mess with these rules!") Today we would simply recommend the standard course of repentance -- avoid the sin, regret it, confess it to G-d, commit to avoiding it again.
An unintentional violation should be fixed by educating yourself and ensuring it doesn't happen again, but doesn't bear any other punishment or guilt. (This came up in the context of a woman who discovered she'd been mistakenly counting 6 clean days instead of 7.)
The first Tosfos on 13b quotes Rabbeinu Tam as saying that she went to the Mikva after 7 days to be Tahor biblically, and then again after 7 clean days for the Rabbinic prohibition. This is why the student thought it was permitted. Either way, everything listed there is a Rabbinic prohibition, and his result was "he died at half his years."
(As a side point, one reason for why she went to the Mikva twice I have seen suggested that the first time was so that she could deal with Taharos - she could make food that was a Tahor, and the second time was to satisfy the Rabbinic requirement for marital relations).