This may or my not be outlined in your cardholder agreement...
I am quite certain that they would NOT incur interest, since points in general are not "money", but a non-monetary benefit which could be redeemed for money. I suspect that instead, you would just "pay back" the negative balance as you earn points with future purchases.
A negative reward points balance means I owe the credit card company some money.
No, you don't "owe" the bank anything, just like they don't "owe" you anything when you earn points - it can be redeemed for cash or goods, but it's not a financial debt to them, and they can "reverse" points for a number of reasons (usually involving fraud or misuse). A negative points balance would just a debit balance that will likely get "paid back" by future benefits that you would have earned.
Should I make some purchases with this card so that the reward balance becomes positive or zero
No - just like you shouldn't make purchases just to earn points, you shouldn't make purchases just to erase the negative points balance. Let is correct itself naturally.
If you want to use it to spend on things that you would have bought anyway, that's fine, but I wouldn't go on a "shopping spree" just to get rid of the deficit, and I certainly wouldn't spend more than you could absolutely pay back without accruing any interest.