Drill the screw and spray the screw with PB Blaster and apply light heat (hand held propane torch judiciously applied) to draw the spray into the threads. Use a flat concrete "cut nail" as an EZ out. The cut nail is soft and will twist (won't snap off like an EZ out) if the screw is stubborn. May need repeated PB Blaster and heat but they usually come out. Use a dab of anti-seize on the new screw threads.

It probably the end of the cartridge. They can be replaced, and Delta can walk you through it over the phone.

Replace the Spigot; the seal in the diverter has gone bad. Call Delta Customer Service they may send you a new one. You will need the Model number and finish color of the Spigot. You can also purchase one at a Big Box store. Check to see if the spout screws on or slips on the Copper pipe. Delta can also tell you this piece of information

Delta and Moen are almost interchangeable in terms of warranty and service. Both offer lifetime warranties on the parts. I'm not sure about Moen but Delta also offers a lifetime warranty on their finishes. If your finish has changed you might want to contact Delta and see if those parts would be covered under warranty. Of course it'd still be a hassle to uninstall and reinstall so if the finish is still fine no need to do anything. We have a lot of experience selling Delta Shower Faucets. You can read our article which will teach you everything you wanted to know (and probably more) about Delta Showers here. In regards to the plastic elements, we get asked about that a lot. We are an authorized Delta reseller and many people ask us about whether or not the shower faucets we sell are metal or plastic. The answer is, usually there's a mixture. Some Delta Showers are all metal, others are a combination of metal and plastic. It simply depends on what shape and style they are going for and how the engineers decided how best to fabricate the part. That being said, keep in mind that even the plastic parts made by Delta have a lifetime warranty. You can literally call them up and they will replace a part that is no longer working. I don't think there's nearly as much to worry about regarding plastic parts as people think. Especially if those parts are backed by a huge company like Delta Faucet with a lifetime warranty. Additionally, people don't always think about the reason plastic has been chosen. In the case of a handheld shower, often plastic is the material of choice because of thermal transfer issues. Metal heats up. If you hold hot metal in your hand it's not so comfortable. Plus, it's possible the hand shower would slip out of your hand and you would drop it. If it's plastic, it's very unlikely to damage your tile or hurt you in any way. If it's heavy and made of metal this may not be the case.