Is Thunderbolt 2 to Thunderbolt 3 (type C) possible? Apple released Thunderbolt 3 to Thunderbolt 2 adapter. Is it possible to connect an external device using type C (ie ASUS MB169B+) by connecting the Thunderbolt 3 to Thunderbolt 2 adapter with an Thunderbolt cable into the Mac (this is an older Mac before 2015)?

Can the MacBook (which only supports Thunderbolt 2) fully access the device which supports type C? The transmission traffic still won't compare with the real Thunderbolt 3 speeds but at least the Thunderbolt 2 ports are not useless!
Let's break your question into two questions.

Can I use the ASUS MB169B+ with a Thunderbolt 1/2 Mac?
Is there a generic way to connect any arbitrary Thunderbolt 3 or USB Type C peripheral to a Thunderbolt 1/2 Mac?

First let me address your specific example. The ASUS MB169B+ is actually just a regular USB 3.0 device. It uses a technology called DisplayLink which basically is a video card connected over USB (instead of the typical PCIe connection.)
Because MB169B+ is a regular USB device, you can simply connect it to one of your MacBook's regular USB ports and it will work just as well as it would connected to a USB Type C port.
Now for the second question, can you use a Thunderbolt 3 (or USB Type C) peripheral with a Thunderbolt 2 computer? Theoretically, yes it is possible to create a dongle that accomplishes this. You would have to put a Falcon Ridge controller and an Alpine Ridge controller in to the same dongle and connect them together like so:
            +--------Thunderbolt 2 to 3 Adapter Box--------+
            |                +--&gt; PCIe --+                 |
MacBook --&gt; | Falcon Ridge --|           |--&gt; Alpine Ridge | --&gt; Thunderbolt 3/Type C
            |                +--&gt; DP ----+                 |
            +----------------------------------------------+

While a dongle exactly as above does not exist, there is an interesting workaround. The Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter you mention has some limited ability here. While the Apple adapter does work as a generic 3 to 2 adapter, it can only go from 2 to 3 if the peripheral on the other side is a native Thunderbolt 3 device. For example, this adapter works for connecting a LG UltraFine 5K Display to a Thunderbolt 2 Mac. However, it won't work with the 2016 21.5&quot; LG UltraFine 4K Display since that monitor is USB-C only and not a Thunderbolt 3 device. It will work with the 2019 23.5&quot; LG UltraFine 4K Display since that monitor has Thunderbolt 3.
Now, here is where things start to get weird. If you buy a Thunderbolt 3 dock, for example the CalDigit TS3 Plus Dock you can use the Apple adapter to connect a Thunderbolt 2 Mac to it. Most Thunderbolt docks have two Thunderbolt ports on them, one upstream port to connect the computer to the dock and one downstream port for daisy chaining more Thunderbolt devices. That downstream port is actually a full featured USB-C port as well, with support for Thunderbolt 3, Display Port Alternate mode, and USB 3.1. So you can actually plug the 2016 21.5&quot; LG UltraFine 4K Display in to that downstream port and it will work!
It's pretty crazy but the following connection sequence actually works:
Thunderbolt 2 Mac --&gt; Thunderbolt 3 to 2 Adapter --&gt; Thunderbolt 3 Dock --&gt; 2016 21.5&quot; LG 4K
But this sequence (un-intuitively) does NOT work:
Thunderbolt 2 Mac --&gt; Thunderbolt 3 to 2 Adapter --&gt; 2016 21.5&quot; LG 4K
So in summary, the combination of the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with a Thunderbolt 3 dock creates a fully featured (and expensive) Thunderbolt 2 to Thunderbolt 3 adapter.