4.4.4 patched the SSL/Hearbleed bug, so you're safe from that. (But if you need to be safe from that on a wireless connection, you're not practicing safe internet.)
Are you safe from everything? Not if you're connected to the internet in any way. (Some day that'll apply to the power grid too - someone will invent a virus that you can send to refrigerators to make them too warm, or something.)
Run
Malwarebytes Anti-Malware and an anti-virus app and that should catch anything trying to play with your phone.
As for the stock browser, Internet ... well, it matches most Samsung apps - calling it barely usable is a compliment. I run
Atlas Web Browser when I want a fast browser (almost nothing there, but it's fast) and
Firefox Browser for Android when I want power and addons.