Is the slaughter of an animal for a wealthy or influential guest, with the intention of pleasing and honoring him to fulfill a worldly need, without the intention of seeking proximity to God, but which he would not have slaughtered for a poor person, considered lawful or unlawful? And does it fall under the ruling of polytheism (shirk) or slaughtering for other than God?

The slaughter that is forbidden by Islamic law is that which is an act of worship to other than Allah, such as venerating jinn. This is polytheism (shirk (definition)). As for slaughtering to honor a guest, it is permissible and has been encouraged by scholars, who advise the host to choose the best available for their guest. It is preferable for the host to spend their wealth seeking the countenance of Allah, not to achieve a worldly need or to receive a better compensation. While the latter is permissible, it carries no reward, nor does it entail sin.