Which perks should I pick?
Choose a weapon skill such as one-handed and/or archery. Focus on perks which give you additional damage. You may also want to invest perks in alchemy (for poisons such as Paralysis and Slow, plus restorative or resistance potions), sneak (essential for an assassin) and lockpicking (to make looting valuable treasures easier as a thief).
The critical multipliers from attacking an unaware foe are the hallmark of an assassin, so increasing base damage (archery/one-handed) and the damage multiplier (sneak) should be one of your priorities.
Should I enchant weapons or use grindstone?
You can do both. If you use the grindstone before enchanting weapons, you shouldn't have a problem. There is also a smithing perk that allows you to improve already enchanted weapons.
Overall, I found that smithing was more useful than enchanting (smithing lets you get the very best weapons and armour much earlier than usual), but a Paralysis for One Second enchantment on my bow was effective in causing enemies to fall over and have to stand up again, at which point they've probably been paralysed for a second time.
Also, bear in mind that you are unable to use the shout Elemental Fury on an enchanted weapon. Elemental Fury makes a weapon attack very, very quickly, so in some cases you might be better off with a non-enchanted weapon.
How would I regenerate health quicker without resorting to magic (even just out of combat)?
Alchemy. It is very easy and cheap to make a large number of healing potions and consume them whenever you need some extra health. Plus, you gain the benefit of being able to make poisons and apply them to weapons to gain bonus effects on hitting your foes.
Should I be forced to get archer perks to kill dragons?
If you play about 1/3rd of the main quest, you will gain a shout called Dragonrend which forces a dragon to land and stay on the ground. You can repeatedly use this shout, and they will not be able to fly. Dragons tend to land regularly enough on their own so that archery isn't necessary, but do expect to have to spend some time dodging on the ground before they will oblige you.
What advantage has dual wield vs single wield? Should I spend many points in one handed weapons tree?
Dual wielding weapons is only slightly better in some situations than using single-handed weapons. You will attack with dual weapons together exactly as fast as you would using a single weapon by itself, until later in the game when you get the perks that increase the attack speed to 1.35 times the attack speed of a single weapon.
Dual weapons prevent you from blocking, and remove the possibility of having anything else in the off hand. If you have a lot of stamina, there is the possibility of using a very fast power attack which deals a lot of damage, but it drains stamina to zero very quickly. I would recommend that you focus on the other options in the one-handed weapons tree or other skills instead.