does a cut gem or a man-made one have the same mystical properties as a natural stone?
It does and it doesn't. If you take a mined stone that was created umptyscrunch millions of years ago it has the connection to the Elements, the Earth Mother, or whatever powers that be that are involved with rocks. Because of that connection it is already attuned for the purposes that it is going to be for.
A raw lab stone is a blank slate. It needs to be taught its purpose and be introduced to the powers that be. Working in rituals alongside mined stones of the same type are one way, doing rituals with the sole purpose of saying "This is a new [stone] and it needs your guidance as much as a stone created by your hand" is another. A third way is to leave it with other stones that have been in service for a while. They will pass on their knowledge.
A funny thing (going on the subject of how stones can communicate), I also have "trained" my stones by storing them with other crystals of the same type. I put a lab grown chunk of sapphire next to a mined one and asked if it would mind passing on its wisdom to the "new kid". I could feel the mined sapphire grumbling and grousing about it.
I apologized and started to take the newbie away. The mined one then started really hollering at me until I brought it back. I did and it went back to grumbling. I tried again and it started hollering again. So I put it back and left it there with the grumpy stone.
I was as if it was saying "mumbly, grumble, why do *I* get stuck with all the rookies? Stupid kids can't even blow their noses without mommy handing them a tissue. HEY! Bring that kid back here. How the @#$% do you expect me to teach him anything if he's not here to be taught!?!? mumbly-grumble stupid human. Boy you better make this worth my time! I don't teach just anyone you know."
I have a feeling that one from a jewellers is too man made and will lose some of its power?
Not lose power. They will pick up other energies that you'll need to cleanse before use, but you'd have to do that with any stone or tool that you pick up anyway.
And remember that your relationship with the stones is a personal one. You and I could look at two stones. One may feel right to you but feel off to me and vice versa. So while I have had good luck with man-made stones, not everyone else will.
The cheap rough bit of quartz I have seems to have a certain magic about it even though it doesn't look so hot.
Looks are not everything. I have some rough chunks of quartz that look like they were right next to the TNT used to blow them out of the rock face. Powerful suckers. I also have a pristine chunk of Herkemer Diamond (a really REALLY clear form of quartz) that is butt worthless. It's priss and just does not want to do anything more than sit there looking pretty. It's like a beauty pageant contestant
"I just want to say that being chosen as this month's Miss August is
like a compliment I'll remember for as long as I can."
But then I've also had pretty stones who were real hard workers and ugly ones that were just lumps. So as with everything, there are differences in personality.