@Svartepjotr: A general take on using different redistribution terms: Yes you may distribute the source and binaries even if you choose follow the restrictions of GPL, but the distributed source/binary still keeps the permission to distribute it further BSD-style. You may not remove that permission - you must supply the terms of the permission given by the BSD license along with the data which you distribute. So yes you can apply GPL to your behavior, it is fully OK but you can not remove the rights granted by the original license, so your recipient will not be bound by the rules you choose for yourself. Thus far, adding a GPL clause to otherwise unmodified BSD source is plainly illegal (as it would be a false statement about GPL restrictions, which actually do not apply to your recipient).
The problem is, you probably do not indicate to the recipient which parts of a modified work are covered by your copyright and thus are not covered by the original license. This in practice makes it impossible to use the original free permission for the parts still being under the original copyright and license. This forces the recipients to obey GPL for the whole, to make use of your added changes.
This is not any "relicensing", just two separate licenses valid for parts of the work, which are usually deliberately mixed together to make them practically inseparable (unless you specify which parts you changed and how, iow distribute your patch to the original code, such a patch would be indeed a clean GPL work).
To answer your question:
yes you can use BSD-licensed code in a GPL-licensed project
you can not "relicense" but you can add your more restrictive license to modified files and this will in practice void the rights granted to the BSD-licensed part, which part will still exist and bear the original license and permissions
you must of course keep the original copyright notice and the license text