Will I need grass and weed killer if I plan to install bedding fabric and then covered by mulch in beds I am creating. I am a little nervous it will hurt the bushes and trees that are already there?

Depends on how lazy you are... weeds will root and grow in the mulch... but they are easy to pull out as they are blocked from rooting through the fabric into the dirt below. Particularly as the Mulch further decomposes

Spraying weed killer close to desired plants can have unintended and undesired results.

Skip the weed killers... you don't even have to 'pull' the few weeds that will grow in the mulch; you can just rake the mulch around enough to remove them.
I walked across a lawn once in my bare feet - my feet swelled up and hurt! All from weed killer. I don't want a world where I can't walk in bare feet, and what about pets? And the ground water?