You couldn't find is_processing() in the timer docs because it's a method of Node. Honestly I don't know why is_active() doesn't normally work, that's kind of a bug.
If you move this stuff over to _input() instead of _process() that will solve the multiple press issue. _input(event) only fires when something actually happens. And then you might not need the is_processing() check either. This should work:
func _ready():
set_process_input(true)
func _input(event):
if event.is_action_pressed("key_space"):
timer.set_active(false)
elif event.is_action_released("key_space"):
timer.set_active(true)