I have a notice period of 1 month for house internet. The last bill covers until 28th June (payment always 27th each month). Today is 31st May. If I cancel my subscription now/today, does it mean that I have to pay for the whole month of July? 

Agreements have a cancellation period of one (1) month. This means that if you cancel now, your subscription will be terminated next month.

You still owe costs for the remaining period, if too much has been paid, you will see this on your final invoice. You will of course get this back on the bank account known to KPN, if you still have to pay for consumption costs (for ex. because you made calls on your fixed line), this will also be stated on your final invoice and this has to be paid (or will be deducted from the credit if paid to much in advance).