Is it possible to connect a 1080p Thunderbolt 3 display to a Thunderbolt 2 port? I have a Thunderbolt 3 device that is a 1080p LCD monitor. When the device is connected, System Profiler lists its monitor as using a DisplayPort connection, as well as a few USB 2 devices since this device has touch screen capabilities. It has no external power source -- it's typically powered over its Thunderbolt 3 connection when connected to an actual Thunderbolt 3 port.

I'm wondering if it's possible for me to connect this device to a Mac with a Thunderbolt 2 port and use it as a monitor? In theory this is possible since Thunderbolt 2 supports DisplayPort and could provide more than enough bandwidth and power for a 1080p LCD monitor. It would be nice if this device's USB 2 capabilities worked as well, but not a big deal if they don't. Basically, I'm not sure any such adapter exists for this sort of thing, or if it would work for my particular needs.

Is this possible? For example, can I use the Apple Thunderbolt 2 to Thunderbolt 3 adapter + a male-to-male Thunderbolt 2 cable to accomplish this? Or maybe just a male-to-male Mini DisplayPort cable?
No, that is not possible. Thunderbolt 3 is required to support the power delivery (USB-PD) over the Thunderbolt cable.

As your display has no external power source, it won't help that you get the display signal to it, as it would have no power.