Does demanding one's full rights contradict Allah's command to be patient? And is it permissible to leave one's current job for another after performing Istikhara regarding the first job, especially with the emergence of problems in the current company?

Islamic law commands seeking what benefits a Muslim in their religious and worldly affairs. The strong believer is better and more beloved to Allah than the weak believer, as stated in the Hadith (definition): "Be keen on what benefits you, and seek help from Allah and do not be helpless." Your demanding your rights from the company or transferring to a more beneficial company falls within being keen on permissible benefit and striving to acquire sustenance, and there is no objection to Allah's decree in that. Istikhara in such matters is legislated.