Is it permissible to ask Allah for things that may seem trivial, such as assistance in walking between rooms or drinking water, despite His عظمه (greatness)?

There is no harm in supplicating for a small matter, even asking for a shoelace if it breaks, because Allah is the One who provides for all the needs of His servant, and He loves to be asked and requested from.

However, a person should not limit their aspirations to these worldly requests. Instead, they should combine in their supplications the good of both this world and the Hereafter, because Allah condemned those who ask Him only for matters of their worldly life and turn away from their Hereafter.

As for seeking assistance (Isti'anah), it is permissible in every action, such as saying "Bismillah" (in the name of Allah) before drinking or walking. This is seeking blessing in the name of Allah and seeking His assistance.