Android x86 Virtual box with Internet and connection to adb?
I'm using the very same setup with NAT mode. To connect via ADB:


in VirtualBox Manager, right-click on your VM and select "show log"
press Ctrl-F to initialize a search, type "DHCP". You will find several lines stating something like NAT: DHCP offered IP address 10.0.2.15 – that's your IP, write it down ;)
select the "gear icon" in the same context menu to "change settings"
go to the "network" tab, and click the "port forwarding" button
add two rules to forward ports 5555 (adb) and 5554 (console) from the guest to your host machine. To make an easy rule for potential multiple Android-VMs, you can simply add a digit to the end – e.g. 5555→55551 and 5554→55541 for this machine (add a "2" for your second, if any).
now on your host machine, execute the command adb connect localhost:55551


Done. Execute any ADB commands as usual. You can use adb disconnect localhost:55551 if it gets in the way (e.g. if you're using some other device, and don't want to always specify which one to use with each ADB command).