Have a look at the description of the stages...normally you see which items can drop...so 6* Items can drop from level 13 dungeons. The probability to get better items rises with the level of the dungeon...but unfortunately the dominant RNG of the game will not guarantee good items...I am sure there is a list with the probabillites...but I do not know where...

6* items start from stage 13 dungeons indeed, but you have a much higher chance at stage 20, which is the only one where 4* doesn't drop anymore, so only 5* or 6*. Also, starting from stage 16, you can no longer get XP brews or mystery shards as a reward, so the chance of getting good gear improves significantly. 

So if you farm dungeons for gear, you shouldn't go any lower than 13 and if you can do 16, it's a big improvement. Ultimately of course you want to be able to do 20. 

If BEK can solo Dragon, adding BEK to a combination of Coldheart and Royal Guard should help me reach 20 , is that correct

Stage 13 onwards it consumes 14 energy and gives chance of getting 4-6 artifacts.

the probability of starting stage 13 to stage 19 is 4-6 artifacts , so is there a table or site which tells us the exact probability or does the probability pretty much the same

BEK can solo dragon with the right build and endgame gear, yes, but for progression to stage 20, you should probably stick with a 5 man team, including BEK of course. Get the right balance of offense, debuffers and support. 

There is a table, yes: https://media.discordapp.net/attachments/631836292031840257/790405540714250300/image0.png?width=960&height=514 

It mentions Spider's Den, but it's valid for Dragon/Golem/FK as well.