I don't know much about the genesis of Redis or Redis Labs, who key people and dates are, etc. I guess this obfuscation is part of the problem.
(Say you like something like Elm -- wouldn't it be better to have a relatively closely aligned commercial entity that puts significant and effective effort in making it widely used, which in turns makes it easier to find an Elm job or sell Elm-like solutions as a consultant).
You grant to Redis and to the recipients of the software distributed by Redis a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable copyright license to reproduce, prepare derivative works of, publicly display, publicly perform, sublicense, and distribute Your Contribution and such derivative works.
Usually, projects don't have such agreements. E.g. Linux, the kernel, would have a hard time re-licensing under different license, because of how many people actually hold the copyright to the code they contributed, and would have to agree beforehand.[0] https://github.com/redis/redis/blob/unstable/CONTRIBUTING.md
Like since Rust is MIT licensed, you could make a closed source fork of Rust. The trademark guidelines would prevent you calling it Rust or anything too close, but you could describe it as Rust(TM) compatible and any of the other legally permitted uses of other people's trademarks.
No, because the Rust trademark guidelines prohibit that (https://foundation.rust-lang.org/policies/logo-policy-and-me...)
In Redis Labs case, they acquired the Redis trademark from the original author, who they employed for a few years after the project was well established.
Nobody could start "Rust Labs" without the agreement of the Rust Foundation, because they own the Rust trademark.
I'm not a lawyer, so take the following with that grain of salt.
In the specific case of Rust, no, because as another user pointed out, their licensing prohibits it.
If my understanding of the licenses is correct, the X-11, BSD 3-clause and BSD 4-clause licenses also prohibit this.
The MIT, BSD 2-clause and ISC licenses don't appear to prohibit this.
Your post mentions a few issues which I believe are legally separate:
1. Naming your company after an open-source project. I believe this is perfectly legal under the latter listed licenses, and happens in practice (for example, a brief search yields that React is MIT-licensed, and "React Labs" is a company).
2. Selling a commercial version of an open-source project. This is legal, and in fact a license isn't considered open source by OSI or free software by FSF if it disallows selling a commercial version. Whether this will be profitable is a separate question--generally people won't be willing to pay for something if they can just get it for free. There are two ways around this that I can think of: a) providing services and development around the open-source project, and b) extending the open source project with closed source code. The latter business model is prohibited by copyleft--you can only sell closed-source extensions to copyleft software if you have rights to the copyright (i.e. you created the code yourself) so attempting to do this with an existing copyleft-licensed project would be prohibited.
3. Enforcing trademark on the name of an open source project. My understanding is that enforcing a trademark created after a open source project started using it isn't possible, not because of licensing terms, but because of "priority of use" or "first to use in commerce"[1]. That is, if an open source project "Foo" exists already, I can't create "Foo Labs" and then sue the Foo project for using my name--on the contrary, the Foo project could probably sue Foo Labs. Redis Labs avoids this liability because they obtained rights for the trademark from the original Redis developer (I'm not sure what terms they obtained rights to the Redis name under--if they have exclusive rights they could sue anyone using the Redis name, but contribution to the project over time would likely make this complicated). There's a separate issue which is that "Foo" and "Foo Labs" are arguably different trademarks--Foo Labs can't inherently sue anyone using the Foo name, but they could likely sue someone who started a "Foo Labs" if they were the first ones to trade under that name.
[1] https://www.avvo.com/legal-answers/does-prior-art-apply-to-t...