There are a few places you can look, but a lot of it will depend on your friend's profession. There are probably several industry-specific sites.
Here are some of the biggest:
LinkedIn is probably the world's biggest job web site.
Blocket Jobs is Sweden's largest job website.*
EURES The European Job Mobility Portal.
Glassdoor Another large, global, job ...
To get a qualified answer you should ask the authorities directly. You could do so on multiple channels via https://www.make-it-in-germany.com/de/ueber-das-portal/kontakt/
I’m pretty sure that you could use both passports for a grace period with regards to the blue card. But you should nevertheless ask.
From my personal experience, usually in case of passport change you can still travel with both passports for some time. Of course, change of surname makes things slightly more complicated. In your case, I would write an email to border control asking if they will let you into the country.
Another option might be that you do the name change and passport ...