I think DTC is just stocks (or bonds) held in "Street Name". DTC used to be a large warehouse in Long Island City, Queens, NY. Owned by NYSE. On the other hand, If you bought 100 shares XYZ and had the physical certificate mailed to your house, that is NOT in "Street Name". But of course you can still sell it, just deliver it to your broker. They will send it to the Transfer Agent. That's from years ago, at least. And that's how Bernie Madoff scammed the regulators. His accounts had nothing in Street Name, but the SEC never even checked. A total BS Ponzi Scheme by Madoff, and the SEC lawyers couldn't figure it out.
No. Retail selling stock certs needs to deliver same to the broker who executed the sell trade. And if you sell 100 XYZ, but your stock cert says 500 XYZ, give the 500 to your broker. The 100 you sold will go to the buyer. You can take delivery of the remaining 400, if you wish. Fees for physical certs apply, but I have no idea how much $$$. That's the way it was before ~ 1990, when Book Entry became the norm. These days, very few people have stock certs. They cost $$$. Too bad. There used to be some really pretty ones, artwork, engraved plates.