What is the ruling on advertising pharmaceuticals in this manner, and what is the ruling on gifts offered to doctors by pharmaceutical companies in exchange for prescribing their products? Is this permissible or forbidden?

It is not permissible for a doctor to accept gifts from pharmaceutical companies, because they are forbidden bribes, even if they are called gifts. Names do not change realities. These gifts lead him to favoritism towards the gifting company, which harms other companies.