Is it permissible for a small shop owner to publish pictures of offers from large competing stores and write a comment that his prices are lower than or equal to theirs, with the name of the other store visible on the pictures? And does the ruling differ if he blurs out the name?

The advertisements presented by stores regarding offers are their private property. Copyrights, innovation rights, and invention rights are protected by Islamic law for their owners, and it is not permissible to infringe upon them. Therefore, it is not permissible to exploit images from these advertisements for negative propaganda and to alienate people from the store that owns the advertisement. This is considered an aggression against a right respected by Islamic law and causes harm to others. You should treat people the way you would like to be treated.