First off, how sturdy and durable are the materials? Particularly the "White Strong & Flexible", "White Strong & Flexible Polished", "Frosted Ultra Detail", and the "Alumide".
White strong and flexible is extremely strong, and has a lot of detail. It's suitable for 99% of the custom puzzles that get posted here. I'm pretty sure the polished one loses some of the details from the polishing process, but it's still very strong. Can't speak for the other two, but unless you're thinking of designing a puzzle, you'll probably be best off with WSF.
Can all 4 of the above materials be lubed? I'll be using Traxxas 50k diff oil to do so.
White strong and flexible is very nice lubed. It usually has a very gritty feel at first, but it slowly smooths out over time because the pieces effectively sand themselves. Lubrication speeds up this process significantly.
Do all of the puzzles come as DIY kits? Do any of them come with stickers?
All DIYs. If you want an assembled puzzle, some sellers do small runs of them, but you'll have to pay them a good amount extra. You can buy custom stickers through Oliver Nagy, Dr. Sticker, and others, or cut your own from Cubesmith vinyl.
What is the shipping time like? From time of purchase how long does it usually take for most of the puzzles to arrive?
Usually arrives in just over a week for me. I had one order once which took over two weeks, but they were extremely busy then because of Christmas.