"It weighs about 5 lbs."

Traditionally, words modifying just about anything other than a noun phrase were lumped into the default category: adverbs.

Here, a more analytical approach is to label about as a quantifier modifier, which is obviously its function (if one is in the 'numbers are quantifiers' camp. Those who define numbers as being different from quantifiers on the grounds of precision will find a difficulty with this.)

Collins certainly recognises numbers used before noun phrases as determiners:

seventy

determiner

...

6. a. amounting to seventy: the seventy varieties of fabric.

In short, yes, about is an adverb here. It means approximately, and is used correctly in both your examples.

EDIT: When I say adverb, I mean it modifies the adjective five, not the verb weigh. In English, adverbs can modify not only verbs, but also adjectives and other adverbs. This Oxford entry confirms that it is an adverb indeed.