If you want to ask what something's weight is, you would say (for those examples at least)
"How much does it weigh?"
"What is its weight?"
Because weigh in this form is a verb meaning "have a specified weight." you need to use "how much does" as the object is the thing performing the action of weighing something. Weight is a noun meaning "a body's relative mass or the quantity of matter contained by it, giving rise to a downward force; the heaviness of a person or thing." and is usually considered a property of the object, which is why you'd use the possessive for that variant of the sentence. You could say
"What weight is it?"
but that's kind of awkward wording and not something a native would usually say. Note also that there is the adjective "weighty" meaning "weighing a great deal; heavy" so you could theoretically say
"How weighty is it?"
but that usually has the connotation that the thing is probably heavy to begin with.
As a caregiver, the best way for you to ask this question would either be
"How much do you weigh?"
"How heavy are you?"
Note, though, that the second one may be considered offensive by some people as, again, it carries connotations that you're considering them heavy to begin with.
Edit: As M.M pointed out in the comments, you also have the simpler question "How heavy is it?" which is more commonly used.