What does mimo actually mean: it refers to different RX/TX antenna chains, and spatial streams.
My Question is can I take advantage of these new technologies with my existing devices that are Wi-Fi 4(2.4ghz) and some of them support Wi-Fi 5.
~ it depends on the adapters/cards in each device. If they are 802.11N and 802.11ac devices, yes I would say you will get better performance then a wireless-n router with a 1x1:1 configuration.
Can these non mu-mimo devices take advantage of the mu-mimo router. Can the router communicate with multiple non mu-mimo devices (Say the router is 3x3 mu-mimi can it talk with 3 non mu-mimo devices parallely?)
~Yes, they will be able to communicate. But only devices in the 5.8ghz band using 802.11ac will be able to take advantage of the mu-mimo capabilities. You should use the 2.4ghz band exclusively for legacy devices. (For compatibly)
Beamforming has been around since the 802.11N standard was released, but it wasn't implemented in alot of consumer products at the time. So if you are lucky enough to have cards/adapters and respective antenna chains, sure beamforming will add a significant improvement in reception.
There are two types that I know of, and the manufacturers don't exactly tell you which one it is on the box.
Explicit Beamforming requires the transmitter (in your case, the router) and receiver (the wireless device on your network) to exchange information.
This is a feature that would be best suited for Wireless-AC compatible products.
Implicit Beamforming is simpler and does not require support on both ends of the wireless network.
Also with these older devices, does the beamforming capability of the router have any effect?
~ implicit yes. explicit, no. Generally what I have seen with routers that have 2x2:2 antenna chains the range is significantly improved for every device.
Does the router treat non-MU-MIMO and MU-MIMO devices separately?
This one depends on the router and firmware. Some routers can switch the carrier signal & encoding scheme on the fly. Some can't, and having legacy devices connected with newer standards may affect the overall bandwidth for every device connected. (This doesn't seem to be an issue with any 802.11ac wave-2 routers that I have been exposed to).
Hope that clears it up for you.