Cautiously
No, careless is an adjective. The adverb form is carelessly.
The adverb form is "cautiously" (carefully, tentatively). The related noun is circumspection, meaning caution or prudence.
No, it is an adjective meaning 'suggesting caution' in doing certain things.
No. Caution is a noun. To use the -LY suffix, add it to the ADJECTIVE form (which in this case is cautious) and you will form the adverb (cautiously).
Yes, "gingerly" is an adverb. It is used to describe how an action is done, typically indicating a delicacy or caution in movement.
That is the correct spelling of the adverb "wearily" (as with fatigue or tiredness). The adjective is weary. A similar word is "warily" (with caution or suspicion).
"Gingerly" is an adverb used to describe how an action is performed with great care or caution, often suggesting delicacy or gentleness. It is not considered a figure of speech but rather an adverb that modifies the manner in which an action is carried out.
Err on the side of caution.
The suffix of "caution" is "-tion."
Caution (noun): They were advised to use caution when hiking through bear country. A word of caution was included in the instructions. Caution (verb): Medical professionals caution against taking too much notice of email health warnings. There's a sign urging caution be used when boarding the escalator.
err on the side of caution.