The "cooled" fluid going back into the transmission is what lubricates the geartrain. There's no pump to pull that fluid in. So, no, you're not going to draw fluid up the cooler tube into the transmission. That's why you're pouring fluid down the dipstick tube.
An inline filter before the trans cooler is a reasonable idea. I have not done that; although I've bought the filters. Got too lazy to actually install them. They're the same as "power steering filters"; and they're a reasonable idea, too. Buy them based on the inlet and outlet fitting sizes--typically 5/16 or 3/8 barbed fittings so that rubber hose clamps onto them. Some filters come with hose sections and clamps, others don't. Verify your trans cooler tubing size, or PS return hose size before ordering the filter.
A small, but significant change to my stated procedure: After the vehicle has been jacked-up and properly supported, begin by dropping the cooler tube. Add "temporary" tube so the fluid is directed from the cooler into a drainpan. Start engine, run engine until fluid "sputters" into drainpan. At the point where the pump is beginning to suck air instead of fluid--so the fluid sputters into the drain pan instead of a steady stream--the trans pan is nearly empty. Shut off engine, drop and clean pan as described previously. Saves a lot of mess compared to dropping the pan full of fluid. Also saves needing a transmission pan drainplug, which I now omit. The drainplug is just another leak-point if you use the cooler tube and running engine to empty the pan.