You spelled "Draven" wrong

The last comment is the correct one, Vex and Delvin give out sidequests (jobs) all over Skyrim. There are Four main capitals you want to worry about, Solitude, Windhelm, Whiterun, and Markarth. Go to Vex and Delvin and choose any job you prefer but you only need to complete Five in each of those Four cities.
Each time you have done Five in any one of those particular cities Delvin will spawn a special mission for you in that city to restore the Thieves Guilds honor there. Do that mission and stop accepting mission to that city while concentrating on the remaining cities, 'til you have completed all Four of the special missions.
You will then be presented a special ceremony in the Cistern declaring you a Master Thief.
Congrats you have restored the Thieves Guild to its former glory.

I've played through the game twice, and also looked it up again just to make sure.
Finding all the unusual gems is 100% optional and can be done at ANY time, completely unrelated to the quest/achievement you're asking about.

Do the small jobs from delvin/vex (like the ledger editing, pickpoeting etc) until you have done 5 in every major city (Whiterun, Solitude, Markarth, Windhelm, Riften) then do the special person of interest mission in 4 of those 5 (there isnt one in Riften as the guild already has influence there), so it is Erikur in Solitude, that battle-born guy in Whiterun, Endon in Markart and the cruel-sea guy in Windhelm, then you 'restore' the guild. Hope I helped.

find all 24 stones and the crown of barenziah, its a bitch but you get some cool stuff in the ragged flagon if you complete it.

no you dont have to find all the stones

All of them. Just keep doing them till you can't get anymore.

It's random and only the guy who you buy things for the Dark Brotherhood can give them to you so i think you have to do him mission only or only his count