TL084 As Voltage Follower
No, no reason for them to be equal, so long as you observe the maximum (and minimum recommended) total voltage and keep inputs within the common-mode range. 
The minimum common mode range is +/-11V with a +/-15V supply. With supplies +15 and -8 you will have a common mode range of +11/-4. The input will work in this range, in the follower configuration. 
Certainly not. The current is guaranteed to be less than 60mA. It's a minimum of only 10mA and that's with a 15V supply. Use the output swing numbers. With a +/-15V supply it can supply +/-10V to a 2K load (5mA). It might be a bit less with a 22V supply compared to 30V, but probably in that range. 
Voltage followers are prone to oscillation if the load is capacitive, otherwise this amplifier is compensated to be unity-gain (the way you are using it) stable.