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I would wait until the Chase is a year old, and then app for a bank card.
You already have a good mix of credit, and I would hold out for a bank card.
As an aside, are you opted in or out?
Being in a building stage, you may want to be opted-in, as any offer you receive based on a CRA listing must be honored if you app and continue to meet the criteria they specified in getting the listing used to make you the offer.
I agree with Robert - time is your friend here. Get at least 6 months of payments on the chase card before applying for anything else.
Work on getting the collection cleared. PFD if its unpaid, GW if its paid.
Yeah, a DV now would be untimely and pointless. Try different contact methods for the PFD offer. Try email contacts and written letters. If you've been asking for the PFD 'in writing' that may be a sticking point. Some CA's are averse to putting such agreements in writing, because such written agreements could potentially damage their relationships with the CRA's.
Stop apping. With a short history and tons of INQs, you won't have much success at this point. You can definitely garden with two cards, since UTIL and payment history have a huge effect on your scores. With two cards, make sure one posts with a zero balance and the other posts with a balance of less than 10%. It may be easier to pay Chase off before the statement date and letting a balance post on Amazon, since the Amazon CL is higher. However, do whatever makes the most sense to you. Do this for 6 months and then see where your scores are at before apping again.
As for the collection, you might also consider paying/settling and making GW removal attempts.