Would connecting an SSD to my Mac mini through Thunderbolt 4 and leaving it connected all the time cause any damage?

Regarding installing the macOS on an external SSD . . . I have done precisely that on my new base model M2 mini.

However, in my case, it's a bog standard Crucial 1TB connected via USB 3.0

USB 3.0 is only a fraction the speed of the internal drive so you would expect a real performance hit but that is not so.

Using bench tests such as GeekBench, I get identical figures whether I am booted from the internal or external SSD.

More importantly, I get identical performance with normal apps including video editing with Final Cut Pro and DaVinci Resolve . . . rendering and export times being identical.

So why go to the trouble of booting from the external? The answer is simple. SSDs have a finite TBW lifespan and the smaller the SSD, the shorter its theoretical life will be.

Booting from the external means that all the wear and tear takes place on that drive which is extremely cheap and easy to replace unlike the built-in one.

The fact that it is also 4 times bigger means that it should last longer in the first place but even if it doesn't, it's cheap as chips to get another and plug it in!

If the mini SSD fails outside warranty, you are left with a very expensive but useless paperweight.

Generally speaking, an always connected external drive will not damage the Mac. Of course, any external drive with questionable build quality is a potential problem. I cannot speak to the quality of the Acasis enclosure that you mention in your post.

Re: installing macOS on an external drive... it's an option, and a good one for many older Macs. I don't see any particular advantage to doing that with the new Mac mini... unless the internal drive is low capacity and you expect to outgrow the space available very soon. As long as the external drive is one of quality and the connection is Thunderbolt or USB-C the experience should be just fine... at least near-equal to running from the internal startup drive.

I don't think it will cause any problems. It is always a good idea to visit the vendor's web site to specifically read their information concerning compatibility with the Mac you are connecting to.

I use the same enclosure on an Intel Mac and one on my Mac M1 and no issues so I would not expect any issues with your Mini.

I have a 4TB Crucial X9 Pro for Mac with photos I transferred from a Drobo Drive. Plugged the SSD into the USB-C on the Mini and the X9 doesn't appear anywhere. Is there something else I need to do?

I use Crucial X8 SSD as my Time Machine backup drive. One is connected to the USB-C port on my M2 Mac mini Pro and is rock solid. Had a different X8 SSD on my M1 Mac mini for two years and it was entirely reliable. These Mac minis are powered on for 16 hours per day, every day, and no external SSD drive issues to date.

I can't speak for the Acasis enclosure, but I have a 1TB OWC Envoy Pro SX Thunderbolt drive connected 24/7 and is used as a boot drive.

I have seen reviews of some DIY enclosures that claim to have destroyed ports. However, I would surmise that the root cause may not have been the drive at all but the cheap cable that was supplied with the drives. I would suggest getting a quality and certified Thunderbolt3 or Thunderbolt4/USB4 cable and toss any supplied by the vendor.