Rookie-Mistake:
I didn't include "native-vlan" - after including this, it worked.
If anyone has a similar Setup to do, feel free to use it:
Client01:
set version 12.3R12.4
set system host-name Client-01
set interfaces ge-0/0/0 unit 0 family ethernet-switching port-mode access
set interfaces ge-0/0/0 unit 0 family ethernet-switching vlan members vl-10
set interfaces vlan unit 10 family inet address 10.10.10.1/24
set vlans vl-10 vlan-id 10
set vlans vl-10 l3-interface vlan.10
EX3300-1:
set version 12.3R12.4
set system host-name EX3300-1
set chassis alarm management-ethernet link-down ignore
set interfaces ge-0/0/0 unit 0 family ethernet-switching port-mode access
set interfaces ge-0/0/0 unit 0 family ethernet-switching vlan members vl-3001
set interfaces ge-0/1/0 unit 0 family ethernet-switching port-mode trunk
set interfaces ge-0/1/0 unit 0 family ethernet-switching vlan members all
set protocols lldp interface all
set ethernet-switching-options dot1q-tunneling ether-type 0x8100
set vlans vl-10 vlan-id 10
set vlans vl-11 vlan-id 11
set vlans vl-12 vlan-id 12
set vlans vl-20 vlan-id 20
set vlans vl-22 vlan-id 22
set vlans vl-3001 vlan-id 3001
set vlans vl-3001 dot1q-tunneling customer-vlans 10-12
set vlans vl-3001 dot1q-tunneling customer-vlans 20
set vlans vl-3001 dot1q-tunneling customer-vlans 22
set vlans vl-3001 dot1q-tunneling customer-vlans native
EX3300-2:
set version 12.3R12.4
set system host-name EX3300-2
set chassis alarm management-ethernet link-down ignore
set interfaces ge-0/1/0 unit 0 family ethernet-switching port-mode trunk
set interfaces ge-0/1/0 unit 0 family ethernet-switching vlan members vl-3001
set interfaces ge-0/1/1 unit 0 family ethernet-switching port-mode trunk
set interfaces ge-0/1/1 unit 0 family ethernet-switching vlan members vl-3001
set protocols lldp interface all
set vlans vl-3001 vlan-id 3001
EX3300-3:
set version 12.3R12.4
set system host-name EX3300-3
set chassis alarm management-ethernet link-down ignore
set interfaces ge-0/0/0 unit 0 family ethernet-switching port-mode access
set interfaces ge-0/0/0 unit 0 family ethernet-switching vlan members vl-3001
set interfaces ge-0/1/0 unit 0 family ethernet-switching port-mode trunk
set interfaces ge-0/1/0 unit 0 family ethernet-switching vlan members all
set protocols lldp interface all
set ethernet-switching-options dot1q-tunneling ether-type 0x8100
set vlans vl-10 vlan-id 10
set vlans vl-11 vlan-id 11
set vlans vl-12 vlan-id 12
set vlans vl-20 vlan-id 20
set vlans vl-22 vlan-id 22
set vlans vl-3001 vlan-id 3001
set vlans vl-3001 dot1q-tunneling customer-vlans 10-12
set vlans vl-3001 dot1q-tunneling customer-vlans 20
set vlans vl-3001 dot1q-tunneling customer-vlans 22
set vlans vl-3001 dot1q-tunneling customer-vlans native
Client02:
set version 12.3R12.4
set system host-name Client-02
set interfaces ge-0/0/0 unit 0 family ethernet-switching port-mode access
set interfaces ge-0/0/0 unit 0 family ethernet-switching vlan members vl-10
set interfaces vlan unit 10 family inet address 10.10.10.2/24
set vlans vl-10 vlan-id 10
set vlans vl-10 l3-interface vlan.10