Looking for in-universe, canonical explanations; not armchair speculation and theories. If none (currently) exist, legends explanation would suffice. Regardless, please provide references and context.
From the way it was played out in The Empire Strikes Back it seems that nobody really trusts the carbon freezing of living creatures and even Boba Fett clearly expresses his concerns to Darth Vader when Han Solo is used as a test subject for this procedure.
Current canon doesn't say much at all about freezing people in carbonite. But, since we can include legends, then I will be using the (legends) Wookiepedia entries on “Carbonite” and “Carbonite gun” as sources for all mentions of carbonite freezing throughout this answer.
There are several references to carbonite freezing of sentient beings:
- I-5YQ: Voluntarily. Survived
- Lorn Pavan: Voluntarily. Survived
- Anakin Skywalker: Voluntarily. Survived
- Han Solo: Involuntarily. Survived
- Savage Opress: Involuntarily. Survived
- Sintas Fel: Involuntarily. Suvived
- Antares Draco: Involuntarily. Survived
- Numerous mostly un-named individuals (some Jedi) by Natasi Daala: Involuntarily. Survived
In fact, what's difficult to find is a reference to somebody actually dying from carbonite freezing. I'm sure that it happened somewhere, but I found no such reference.
Many people, especially those frozen involuntarily, did report moderate to severe temporary side effects, but even people who are frozen by the most uncontrolled and/or wacky use of carbonite seem to always survive.
The Empire Strikes Back dialogue doesn’t make much sense if you only consider the history of carbonite freezing. Anakin Skywalker, a.k.a. Darth Vader, is questioning the safety of a procedure that he himself voluntarily underwent and survived?
It only makes sense if you also consider two key points from that situation:
- The carbonite chamber in question is used for one purpose, and that's not freezing organic material. This is mentioned explicitly by Lando.
- Both of the subjects are considered extremely high-value. Vader values Luke and Fett values Solo (also explicit statements.) Even a low probability risk is worth mentioning when such high consequences are associated with any problems.
Therefore, the Empire Strikes Back dialogue seems to reflect concern about an unlikely but disastrous possibility, not a concern about a common possibility.
On to the second part of the question.
Considering the events of The Mandalorian happen five years after the events of Return of the Jedi — which itself take place about a year after the events of The Empire Strikes Back — did news about the usefulness of freezing living creatures in carbonite become common knowledge six years later in the Star Wars universe? Did news of Han Solo’s carbonate captivity in Jabba’s palace get out and suddenly it became a trendy/cool thing for bounty hunters to do?
Carbonite freezing is ancient. It was the successor technology to colony ships, and was actively used as a weapon during the Clone Wars. The earliest reference I can find is 980 BBY, when a carbonite gun is considered for use in subduing Darth Bane. It's been around in portable, weaponized form for over a millenium by the time of your question's setting.
There aren’t any references to bounty hunters specifically using carbonite, but it’s apparent that carbonite is commonplace. A carbonite insert is even a common component of a hyperdrive.
Therefore, it is not surprising in any way that bounty hunters might be using carbonite if they find it convenient.