1. You can enter into an agreement in writing before solemnisation of marriage that no dowry amount was demanded/taken by your side from their side.
You can also mention that no other articles too were demanded from their side.
This agreement can be witnessed by two persons, one from your side and another from their side, the agreement drawn along with any other conditions as per prevailing circumstances, may be prepared in duplicate, with each side retaining one copy.
This may come to your rescue in case of the future issues you apprehend now.
2. Whether or not she cannot claim any property standing on your name of your parents' name, there is no provision in law for that.
3. The answer to this question is also answered as above.