Let me describe a hypothetical scenario, I wish to know if my reasoning is correct.
I use source code X, licensed by author Y under the BSD-3 license, in my source code. I intend to adhere to their BSD-3 license.
Then I sell my product under my custom closed source license. My product does not include any source code. My product's license does not contain any part of Y's license. My product has a warranty and that is stated in my license. Y's copyright notice, the list of conditions, and the disclaimer (i.e. the entire text of their BSD-3 license), is included in the every copy of documentation of my product, as per the second condition of BSD-3.
Are those two statements valid while at the same time not violating Y's license:
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My product is licensed under my license only.
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The disclaimer (written in UPPERCASE) of BSD-3 license is only relevant so that it absolves author Y of any liability, and doesn't have any effect on my license.
Best Answer
1) Yes, if you put license L1 on your program P1, then P1 is licensed under L1, and that's it.
However, it is possible that you are not legally allowed to use license L1. In particular, if one of your dependencies has a license L2, it may state that L1 must meet some specific conditions, or perhaps that L1 must be L2 and nothing else. When people talk about licenses being "compatible" with each other, this is the sort of issue they're talking about. In this case, L2 is the BSD-3 license, which imposes no restrictions at all on what license you use for L1.
In practice, assuming your license is super-restrictive, this means that anyone who downloads your program may not take out that dependency from your program and use it in their own work. They would have to go get that dependency from a distribution with a more permissive license before they could use it in their own work.
2) Yes, the disclaimer is merely saying that you cannot sue the library author for anything. It's not imposing any additional conditions on what you can do with the software, much less what license you may use for derivative works.
P.S. Like everyone else is saying, you need to include a full copy of the BSD-3 license in your distribution of your program, no matter what license you choose for your program. I think the other answers are assuming that "documentation" is not distributed with the program and thus including the license in your documentation isn't good enough; hopefully you meant documentation that comes with the program.