If you install a open-source ROM on your android. Can still an invisible rootkit (from manufacturer) monitor your tweets Invisible means being invisible even in USB connection
Yes. The manufacturer can do anything they like to the phone. There could be spyware in the radio firmware, which monitors transmitted packets. There could be a filter in the bootloader, which modifies any system image you flash to add spyware. There could be spyware in any of the closed-source drivers, which would be the same in your custom ROM as in the official image.
If you don't trust the manufacturer, don't buy their phone, but on the whole, a phone OEM monitoring your tweets is not a likely threat.