All sources should connect directly to your AV receiver.

The receiver should have a video output which connects between the receiver and the projector.

It should also have speaker connections for all the speakers.

FIRST CONNECTIONS:
ROKU->HDMI->RECEIVER

VIDEO OUT
RECEIVER->HDMI->PROJECTOR

AUDIO OUT
RECEIVER->SPEAKER WIRES->SPEAKERS

If you have video dropping after connecting things this way, turn off 'automatic input search' on the receiver and the projector.
If it still happens, change HDMI cables to a different type.
Connect the Roku directly to the projector to see if the issue persists.
Try a different source (cable box/computer) and see if it persists.

The HDMI cables can be a major factor as well as the intermediate item, the receiver. So, it could be an issue with the Sony itself.

If looking for high quality HDMI cables, check out Monoprice.com

Just a quick note: Please provide the make and model of all equipment you are using. The Optoma 1080p Darbee projector, is a mostly useful piece of information, but would be helpful to know the receiver and any other equipment which may be in use.

I typically get cables from Monoprice.

For short interconnects from sources to my AV receiver I use these cables:
https://www.monoprice.com/product?p_id=24187

These are all 4K certified HDMI cables and work up to 18Gb/s HDMI speeds which is what you want for full 4K support.

For the receiver to the projector, things get trickier and can be far more expensive. The going recommendation is Ruipro.

https://www.amazon.com/RUIPRO-4K60HZ-HDMI2-0b-Supports-HDCP2-2/dp/B06XGDFCSC/ref=sr_1_1_sspa

Not exactly cheap at $140.

Monoprice does offer active 4K cables as well, but they are hit or miss at times...
https://www.monoprice.com/product?p_id=12735

Under $35 for it, but reviews are mixed.

Keep in mind these active cables are directional. Watch the ends and make sure you install it properly.

ALL THIS SAID:

Your setup is just 1080p. So you should be able to get away with relatively inexpensive cables which only support 1080p. I have used KableDirekt off of Amazon without any issues out of my Yamaha receiver and my Denon receiver to my projector and TV respectively. I have also used Blue Rigger with the same results...
https://www.amazon.com/BlueRigger-Wall-High-Speed-Cable/dp/B005LQ83EI/ref=sr_1_6

Cheap, works well.

Not sure why, if you are hooking things up as I directed, that you are having any issues getting video to your projector.

Make sure to read the instruction manual for your Sony so you are on the right input of the receiver and that everything is plugged in fully and completely. It is very helpful if the Sony has an on-screen menu to help you out.

It is possible, with a VERY old receiver, that it doesn't support the HDMI you are using, but this is unlikely since you said it has ARC on it.