How to Make All Skill Gem to Level 1

Hi, I'm currently confused with skill gem level system, and here's my confusion written right below.

If I talk to vendors and try to buy some skill gems, some of them are over level 1.

1. Does gem level share among characters?
For example, if I upgraded "Sunder" gem to level 4 as marauder and log in to different character "Templer", does templer has access to level 4 sunder if he tries to buy it through vendor?

2. Is there any other way to reset all the skill gem level to 1?
I know there is a recipe for this, vendor orb of regret and the skill gem to downgrade the gem for only 1 level. Howvever, it takes a lot of currency to downgrade all the gem to level one. Is there any other way?

<Most Important>
3. If I delete a character, all the skill gems reset to level 1?
For example, let's assume that I have leveled up my sunder, CWDT, and other gems to level 4 as marauder. Then I delete the character and make a new marauder. Do i get ALL level 1 skill gem in a new character?

Please help me out with this issues!

1. No, depends on character level.
2. You can just buy a level 1 from lily in act 6. Maybe from Siosa in act 3 too. Can’t remember if his are leveled. You can also trade a level 20 gem plus a gcp for a level 1 gem with 20 quality. Doesn’t work with corrupted gems.
3. No.

Thank you very much for your comment!
It helped me a lot understanding gem level system.