What is bribery, and does it include a public employee accepting free goods from stores over which they have influence, with the owners' consent? And does sin befall those who eat or drink from these goods, such as their family members?

Bribery is money given to establish falsehood or abolish truth. If the shop owner does this with the intent of achieving forbidden things, then it is forbidden bribery, and it is not permissible for his household members to eat from it except out of necessity. However, if the intention is to strengthen relationships and show affection without assisting in falsehood, then it is not considered forbidden bribery.