LCD power requirements

In terms of power and how to power things. The real answer is "It depends".
It depends on the power needed for your device(s) and how the power is being supplied.
For example, the official Arduino Uno uses a wimpy on board power supply and regulator design, particularly on the older Uno boards.
If you power the Uno using an external supply through the barrel connector, it can generate lots of heat for the regulator.
But the amount of heat and how much power it can supply before it fails depends on the overall current draw and the voltage being supplied to the barrel connector.
i.e. using a 12v power supply will generate more heat and be able to supply less power than than a 6-7 volt supply.

If you use the 5v USB connector for your power, you can supply quite a bit more current to attached devices with no regulator heat issues as it by passes the on board regulator.

For example, I do testing of my lcd library with up to seven LCDs with i2c backpacks and an i2c RTC all being powered by the Arduino board.
The Arduino board is powered by USB and the devices pull power from VCC off the Arduino board.

Some of the 3rd party Arduino clones use a better design and do not have as much as an overheating issue when using a external power supply to power the Arduino clone board.

Things can get a little more tricky if using multiple power supplies.

This is why I was wanting to see photos of what you have and how you have it wired up
since these are important details in solving your issues.

My advice would be to try avoid using an external DC supply going through the barrel connector to power the Arduino.
Ideally, if you can, use the 5v USB for powering everything.
Verify your loads using spec/datasheets and/or a meter. If the loads are too high, try to use the 5v USB for the Arduino board and as else much as you can, then bring in the external power (potentially even a different voltage) for the bigger stuff like motors that can be isolated using transistors, relays, etc...

--- bill