A: Don't touch the bees. Or the butterflies. Or really, just avoid any animals. Trust us, it's not worth the risk. Stealth travel techniques might help, but no promises.
A: Don't worry about paradoxes. Worry about the paperwork. Seriously, the paradox prevention department is still working on their budget proposal. In the meantime, just don't kill your grandfather. Or, you know, just don't do anything that might cause a paradox. More on paradox prevention
A: Only if the cat is wearing a tiny fedora and a pair of shades. Otherwise, the cat will probably just get lost in the space-time continuum. More on feline time travel