does mac osx need adobe flash player?

No.

Not to search the web, no.

There are some sites that have content that will require Adobe Flash Player, mostly video. But if you can get by without those, then Flash is not necessary for search or email or most other things.

OS X does not "need" it to run, but everyone's uses will vary and if a site requires Flash to run that you need to use, then you will need to install Flash. For instance, my bank uses a flash-based banking service, so while I don't need Flash for most purposes, in order to use this service I will need to install Flash.