Can I develop a BSD-licensed software that extends an Apache-licensed software (and vice-versa)?
The short answer is that if you use open source software in your project, you must satisfy all the requirements laid out in that license.
Neither BSD nor Apache 2.0 is a "viral" license, meaning AFAIK it makes no demands on the other source code you include it with. That also means that unlike GPL, you aren't releasing a "product" under a BSD or Apache license, but each file or section of code is licensed under the license that the author released it under.
So you could have a project where one module was BSD and one module was Apache, and I see no reason why you couldn't distribute that, as long as you were explicit about it. Note that IANAL.
Do you have the right to take Apache licensed code and re-license it as BSD? I highly doubt it. I believe Apache has a couple more restrictions on it than BSD. You'd have to keep the original copyright notice, of course, but I also believe you have to do a bit extra when releasing changes to it, and you couldn't (or at least shouldn't) remove those conditions by changing the license.
Short answer: Yes. The Apache Software License was based in large part on BSD and MIT style licenses. The common understanding while I was working on Apache code a while back was that you could incorporate BSD, MIT, and similar licensed libraries but you could not do the same for GPL based licenses.
If you have further questions, I highly recommend going directly to Apache:
They are a group of really nice guys, and they are pretty reasonable.
The best answer for this can be found in the apache third party license policy. This is the policy that all apache projects have to follow when using libs with other licenses. As you can see BSD is listed in Category A - Authorized licenses. So it should be no problem using this license. You can also find policies for other licenses here which is quite helpful. In fact I think the Apache policy may also be a good starting point for you own licensing policy if you work for a company.
Here is very useful site: Easily to compare different types of licenses.
In both of them you must:
But in Apache, unlike BSD you must:
This info taken from tldrlegal.com
And as they notice:
Please, see discussion in comments with @DmitryAlexandrov for more info.