Is there a way to lock in both the flash power and exposure time?
Not easily with the 600D in Av exposure mode.
Canon cameras (and most other DSLRs) are programmed to interpret that [flash + Av exposure mode] means the user wishes to use slow shutter sync, often referred to as dragging the shutter. The camera assumes you want the ambient lighting to be primary and the flash to be used as fill. The ambient lighting is regulated by the shutter speed, the flash illumination is regulated by the TTL routine and the user selected Flash Exposure Compensation (FEC).
You can restrict the shutter speed when using Av exposure mode to either 1/200 second fixed or 1/200-1/60 second auto via the Custom Functions in the third Setup (yellow) menu tab---> Custom Functions (C.Fn) ---> C.FN I: Exposure ---> C.Fn-3 Flash sync. speed in Av (Aperture Priority AE) mode. The default setting is Auto which allows a range from 1/200 second (your camera's flash sync speed) to as long as 30 seconds. It's all on page 252 of your EOS 600D Instruction Manual.
Some of the upper-tier EOS camera models with more physical control buttons and customizable mapping of those buttons allow the user to designate different buttons for Exposure Lock (EL) and Flash Exposure Lock (FEL), but none of the xx0D/Rebel or xx00D/Rebel models have such capability.
If you're shooting in low light and want to use the lens at its widest aperture setting you can shoot in P exposure mode. The shutter time will be restricted to 1/60 second at the slowest. If 1/60 second and the lens' widest aperture will still underexpose the ambient light then the flash power will be adjusted to properly expouse the subject (designated by the AF system).
Beyond that to precisely control the shutter time you need to use M exposure mode. E-TTL will still set the flash power automatically based on the user selected FEC and will attempt to properly expose the subject. How the ambient and flash are balanced will be left totally to the aperture, shutter time, and ISO you select.
