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Yes, especially since you have contacted the merchant in good faith and they haven't taken care of the problems.
Yes, since you only have a limited time to dispute a charge, you could fall outside that window if you wait.
You have 60 days from the date the statement generated with this charge on it to dispute it with the bank so if you are not getting satisfaction I would file said dispute prior to this time expiring, you can always cancel it once the merchant follows through. I am likely to be filing a dispute on a furniture company as well for not taking care of a problem with a sofa that I bought a few weeks ago.....
Thanks. I have filed the dispute.
The additional benefits/rights you have depend on the card you have and what benefits it offers. Refer to your guide to benefits.
Thanks. I have Amex Blue Cash Everyday (the one without annual fees). Not sure if I would be able to simultaneously do the same thing. Also, their documentation says that I can file this thing only when merchant is not willing to accept the merchandise. In my case, they are not even responding to my emails.