Is it permissible to go for 'Umrah with non-Muslim relatives who will enter the Sacred Mosque against my will, given that I have no other Mahram? And is that considered a sin that displeases Allah, or is it permissible for me to perform 'Umrah with the intention of thanking Allah?

It is not permissible for a disbeliever to enter the Sacred Precinct (Haram), nor is his Umrah valid. The disbeliever is punished for Hajj, Umrah, and all other secondary obligations of Islam. Judging a specific individual to be a disbeliever is not a simple matter; a person may commit an act of disbelief without actually becoming a disbeliever due to the presence of impediments such as misinterpretation (ta’wil) or ignorance. A ruling of apostasy must be referred to a judge. Ibn Taymiyyah pointed out that many jurists assume that whoever is called a disbeliever is subject to the outward rulings of an apostate, but the correct view is otherwise. For example, hypocrites were treated as Muslims regarding inheritance and inviolability unless clear legal proof was established against them.

As for performing Umrah with the intention of showing gratitude, there is no harm in it. Every act of worship performed by a servant is an expression of gratitude to Allah Almighty for His blessings, and gratitude can be expressed through action, speech, and intention.