Plastic is a word that originally meant “pliable and easily shaped.” It only recently became a name for a category of materials called polymers. The word polymer means “of many parts,” and polymers are made of long chains of molecules. Polymers abound in nature. Cellulose, the material that makes up the cell walls of plants, is a very common natural polymer.
Over the last century and a half humans have learned how to make synthetic polymers, sometimes using natural substances like cellulose, but more often using the plentiful carbon atoms provided by petroleum and other fossil fuels. (Source)
When you say plastic, I assume what you really mean is manufactured polymers. I'm going to intentionally separate that from synthetic polymers, which would be difficult to rationalize and, I suspect, would be unnecessary. To continue...
Natural polymers exist and so, using them, not only is it entirely possible for "plastic" to be available to medieval alchemists — they had them.
There are two types of polymers: synthetic and natural. Synthetic polymers are derived from petroleum oil, and made by scientists and engineers. Examples of synthetic polymers include nylon, polyethylene, polyester, Teflon, and epoxy. Natural polymers occur in nature and can be extracted. They are often water-based. Examples of naturally occurring polymers are silk, wool, DNA, cellulose and proteins. (Source)
From that same source we learn that pectin and, more importantly, rubber, are also natural polymers that could have been available to medieval peoples (with a little help of good ocean-going ships).
It's worth pointing out a bit of "reality" to scope the answer
All technology stands atop a pyramid of knowledge and experience. What you're basically asking us to do is indicate just how much that top of that pyramid can be pulled "back in time." The honest reality is, "not very much" and the further you want it pulled back the less realistic it will be. However, we should be looking for suspension of disbelief, something just "real enough" that it lets you tell a cool story.1 This is important, because of you want manufactured polymers in the medieval era you will be required to ignore how history actually played out and why.
99.99% of all the technology we enjoy today was invented in the last 150 years. You're trying to pull an awful lot against, shall we say, the inertia of innovation. I'm just pointing out that you'll need to be prepared to handwave or ignore things to rationalize what you want. And the further back you want to pull it, the more you'll have to handwave or ignore.
OK, now back to our regularly scheduled program
Here's how the first "plastic" came to be:
It was in 1862 that Alexander Parkes introduced the world’s first-ever man-made plastic, at the London International Exhibition. “Parkesine,” as it was called, was marketed as an alternative to ivory and horn that Parks discovered while trying to develop a synthetic substitute for shellac for waterproofing.
Though the product was not a commercial success, Parkesine represented an important first step in the development of man-made plastic. The material didn’t start to truly show its potential value and diversity of applications until John Wesley Hyatt in Albany, New York discovered a way to manufacture an improved version of Parkesine, most commonly known as celluloid. (Source)
And to expand a little...
The first synthetic polymer was invented in 1869 by John Wesley Hyatt, who was inspired by a New York firm’s offer of $10,000 for anyone who could provide a substitute for ivory. The growing popularity of billiards had put a strain on the supply of natural ivory, obtained through the slaughter of wild elephants. By treating cellulose, derived from cotton fiber, with camphor, Hyatt discovered a plastic that could be crafted into a variety of shapes and made to imitate natural substances like tortoiseshell, horn, linen, and ivory. (Source)
This is a very important stopping point. If Parkesine or celluloid solve your problem, you're in like Flynt.2 The reason is that, being naturally derived, the rationalization for discovering these processes is a whole lot simpler than rationalizing synthetic polymers.
Synthetic polymers didn't really come to be anything like what we know plastics to be today until just before World War II with the development of Nylon (a synthetic silk). I want you to think about all the chemistry that took place between 1862 and 1935. That encompasses the entire Industrial Revolution and it's no small thing to take just a chunk of it out and figure out how to make just that chunk happen 1400 years earlier. Remember what I said about that pyramid.
Consequently, I think I'm going to skip figuring out how to create synthetic plastics, which means that technically I didn't answer your question because I'm not explaining how early medieval Europe could discover and chemically manipulate crude oil more than a millennium before it actually happened. I think that the natural substances will get you to where you want to go (cursing medieval Europe with manufactured polymers long before they were ready...).
I'm also going to completely ignore your second, third, and fourth questions. Please remember that asking more than one question is a reason to vote to close a question (VTC:Needs more focus). You should delete your last paragraph and figure out how to ask it as a single question in a new post. Methinks you'll find there are a lot of potential uses for manufactured polymers in medieval Europe — especially after you've settled on what kind of plastic you'll really have.
1 If you've ever watched "The Murdoch Mysteries" you'll know exactly what I mean. One of the many premises of the show is the "what if?" component of having main character William Murdoch involved with many of the notable inventions and discoveries of the time period. Oh, they take outrageous license with history — but who honking cares? It's fun to see them interweave those discoveries into an alternate history where Murdoch seems to be the magnet for them all. What's the point of this footnote? To remind you to not take being "as close to real life" very seriously. History and science are tools for the writer and worldbuilder — not limitations.
2 "In like Flynt" is a lovely phrase suggesting a flawless and seemingly simple insertion into a complex situation or difficult location. It comes from the movie In Like Flynt (1967) staring the incomparable James Coburn. Most kids nowadays would probably burn the film for its 60's take on male-female relationships (not unlike tearing down statues...), but if you're willing to suck it up and look past, well, the past, you'll have a lot of fun watching the movie.