I'm running OSx Yosemite 10.10.5 - after downloading and successfully updating Flash Player from the Adobe site using Firefox, I noticed that Safari (currently the system's default browser) opened and closed itself briefly. Is that behavior expected?
Also, while using Chrome earlier on, a website complained that my Flash Player was out of date and proceeded to download a .dmg file claiming to be an FP update without asking for confirmation. Are you aware of a way that such a file could unpack and install itself without admin privileges?
The Flash Player installer will not automatically close the browser. If Flash Player requires the browser to be closed to proceed with the installation the user will be prompted to close the browser, via message on the installer dialog window. When the installation completes, the default browser does launch.
What is the link to the website you visited when Chrome complained Flash Player was out of date? If you can reproduce this behaviour, post a screenshot of the website. See How do I attach a screenshot for instructions on taking and attaching a screenshot to your reply. The sequence of events you describe don't appear to be official Flash Player installation workflow. When the user is prompted that Flash Player is out of date, the user is usually prompted to go to get.adobe.com/flashplayer to download the installer file. If a file downloaded automatically without your approval it could be from a malicious site.