Well I can do all three constraints in 6 Turns:
1. Buy Bom
2. Buy Masq (use Baker coin)
3. Bom (Procession) -> Masq (pass Nec, trash Oge, draw 3 C, gain 1 C; pass Hov, draw 2 C, gain 1 c), gain Fortress) : Buy Mint
4. BomMintFortC -> Bom (Chap) -> Trash all
5. Bom (Masq) -> pass Fort, trash whatever
6. Bom (Tmap/Mv/Dthcrt)
Obviously playing Bom as Island in T5 fulfills challenges 1 & 3 in 5 turns. Likewise, gaining anything else in T3 besides Fortress from the Masq fulfills challenges 1&2 in 5 turns.
I can get a 4 turn solution, but only with a second player:
P1 Opens Bish/Bish
P2 Opens Masq/Bom
T3P1: Bish (P2 trashes Oge & draws C)
T3P2: Bom (Prssn) -> Masq (Pass Nec, gain Oge, trash Oge, draw 3 C; pass Hov, gain Nec, trash Nec, draw 2 C, gain Mining Village): buy six C, buy Mint
T4P1: Bish (P2 trashes C)
T4P2: Mv (trash it) -> Bom (Island: set aside Mint).