Will those "mixed alternate mode" adapters work on a Thunderbolt port as well, or are USB and DisplayPort mutually exclusive on Intel's Thunderbolt implementation?
Apple's AV multiport adapter does work with a Thunderbolt 3 port. From this product description:
The USB-C Digital AV Multiport Adapter lets you connect your MacBook with USB-C port or MacBook Pro with Thunderbolt 3 (USB-C) ports to an HDMI display, while also connecting a standard USB device and a USB-C charging cable.
USB-C has dedicated lanes for USB 2.0 signals, so DisplayPort and USB are never mutually exclusive. One dock which does both is the HyperDrive USB Type-C Hub with Mini DisplayPort.
However, I don't believe any USB-C adapter handles DisplayPort and USB 3.1 alternate modes simultaneously. If you want both you need to go full native Thunderbolt 3.
The MSI mainboard and the Razer laptop are both USB HOSTS. The connection between two USB link "partners" can be only if one is HOST, and the other is DEVICE.
The Type-C USB specifications do define so-called DRD - Dual Role Data ports, which can turn itself into being a host or device. However, to have such dual-role functionality from a PC, the system must have TWO USB controllers inside, one acting as xHCI-compliant HOST, and another IP (piece of silicon) that can act as USB device (with all necessary attributes as USB class etc.) The external Type-C port then is internally multiplexed into xHCI or DCI hardware block depending on cable identification rules. The mainstream desktop systems do not have this functionality designed in, but all Intel systems that targeted "mobile space" (Atom line of SoC) usually do have this dual-role functionality.
So the answer to the first question is: yes, you can try to connect both systems with C-C cable, but chances of data transfer are slim, unless the one of systems clearly identifies itself as Dual-Role system (aka "OTG") in its datasheet.
Even if a silicon hardware does have this internal USB capability, the system must provide proper handling of Type-C CC (communication) channel, to determine the role of ports, and engage proper software stack. The Type-C port must be designed in a way that no VBUS power is applied until a cable is plugged in, and both ports sort their identities out. Host ports have CC pull-ups, while device ports have 5.1k pull-down on their CC pin. If both ports have the same "gender role", VBUS power shouldn't be turned on. So the answer to the second question is - if both systems are USB-IF certified, there should be no concern of VBUS power collision.
The answer to the third question is negative, there is no need in any "crossover cable", all crossover is handled in muxes behind the Type-C connector. If both ports are DRP, they will periodically advertise their identity, and the result will be selected at random
Best Answer
USB 3.1 Gen 2 (SuperSpeed+, 10 Gbps) was designed to work over both existing USB 3.0 cables (the ones with the 5 extra contacts), as well as USB Type C cables.
Since existing USB 3.0 cables (the ones with Type A and B connectors, as well as the micro A and B variants) only contain one super-speed pair-of-pairs (Tx pair and Rx pair), USB 3.1 Gen 2 could only use that one pair-of-pairs and still work over existing USB 3.0 cables. So even when you run USB 3.1 Gen 2 over a cable with Type C connectors, it only uses the one super-speed pair-of-pairs. This also makes it possible to have USB 3.0/3.1-capable cables with a Type C connector on one end, and the earlier USB 3.0-style Type A, B, micro A, or micro B connectors on the other end.
Now you might ask a follow-up question, "Why didn't the USB Implementers Forum (USB-IF, the USB standards consortium) define an even-faster-than-10Gbps flavor of the USB protocol, that uses both super-speed pairs-of-pairs in the Type C connector?" That's a valid question, but I'm unwilling to speculate. It would certainly have been a bigger departure from previous USB PHY designs, in that it would have two separate send and receive data streams that would have to be coordinated. In effect, it would be a kind of parallel interface whereas USB has traditionally been nominally serial.
The way you asked your question exposed a few potential misunderstandings that I'd like to address here:
That's not quite true. There are many high quality USB Type C compliant cables that are not suitable for Thunderbolt 3. Thunderbolt 3 is limited to ≤ 0.5m cable lengths if you have a passive cable. To go longer than that (like 2m), you need a more expensive active cable (a cable with special IC chips in it to assist in signal handling).
USB-C is not a protocol. USB Type C is the name of a connector and cabling specification; it's not the name of the protocols that are used over those connectors and cables. When doing USB protocols over Type C cables, you're doing USB 3.1 Gen 2 (10Gbps "SuperSpeed+"), or earlier flavors of USB.