The HDMI connection should be made prior to powering up the Raspberry Pi. Some monitors (and probably some projectors) may also need to be powered on prior to powering up the Raspberry Pi. Make sure you have a good solid connection (cable isn't half-way inserted), and maybe try a different HDMI cable.
As long as the Raspberry Pi's 5v USB power supply is adequate, the BrickPi will run. If it's a BrickPi+, don't try running motors when you don't have batteries connected, and note than some sensors might not work as expected. If it's a BrickPi3, the motors will automatically be disabled if there isn't adequate battery voltage, and all the sensors will work normally. The BrickPi3, even with sensors, uses minimal power from the 5v rail (an order of magnitude less than a Raspberry Pi with WiFi).