I don't know if there are conjunctions that can never precede a participial phrase, but certainly many conjunctions won't work with many phrases. I'd say most such phrases can only be used with one of a few possible conjunctions.
For reference, I'd like to link this question on what makes a participial phrase here.
- Talking with my friends, I always release my stress.
- When I talk with my friends... or when talking with my friend...
Here, (2) sounds natural (though saying "I release my stress" sounds more like you're intentionally choosing to let go of your stress, as opposed to just forgetting about it, being distracted, or being comforted).
Most of the conjunctions you list work with this sentence, though (since they're conjuctions) they need to have something before them.
Talking with my friends, I can relax.
Is a whole phrase and thought. If you want to add conjunctions, you need something before it:
I'm always stressed after work, but talking with my friends, I can relax.
I can think of sentences that could use but, and, while, because, or, as here, but not once.
- Once seeing you, I feel quite guilty.
Here, once doesn't work, and would need to be "seeing you..." or "once I see you...", or "upon seeing you...". An example where once does work could be:
Once soaring across the sky, the aircraft now sat gathering dust in an old hanger.
or (using a different meaning of "once"):
Once flying high above cloud level, the pilot turned off the fasten-seat-belts light and signaled for the attendants to begin passing out the tiny bags of peanuts.
I can't think of very many other examples where 'once' can be used in front of a participial phrase though.
- Although hating to go with his girlfriend for shopping, he still gives in to her every time.
Here I'd just want to mention as a side-note that a more common way to say "to go with his girlfriend for shopping" would be "...to go shopping with his girlfriend...".
Aside from that though, "Although he hates.." sounds more natural to me, but I think
Although hating to go shopping with his girlfriend, he still agrees every time.
isn't actually wrong. It'd certainly be understood.
- Because leading a rural life, I often go other places by bicycle.
Here you'd need to re-phrase the rest of it slightly, but only because you're using the word 'because'. Both of these are fine:
Because I lead a rural life, I often go other places by bicycle.
Because leading a rural life means having little extra money for gas, I often travel by bicycle.
Using a different conjunction though, that re-structuring isn't needed. I have no idea why. These, for example, are fine:
- When leading a rural life, people rarely miss the roar of city traffic.
- While leading a rural life, many Amish people still find technology interesting.
You can remove "when" or "because" without error.
- As listening to the music carefully, you will be impressed by its melody.
This also is a little off, but replacing 'as' with one of a number of other conjunctions (but, and, while, because, after, +some more probably) will make the sentence work.
You may just want to look up examples of conjunctions - I'm a native speaker, and it seems most conjunctions have their own, slightly different sets of rules.