Your 3/4" 5600 is going to reduce your pressure during backwash and you can not increase the length of time the backwash etc. can run for (as Bob mistakenly said you can), so you should not add anything before the AN filter that will reduce the pressure farther, like this prefilter you have no need for.
What you think is sand etc, probably is hardness scale or dirt coming out of the plumbing but either way your AN mineral will filter it out down to about 20 microns. You can't see particles less than 50-45 microns without a microscope.
And IF you have good pressure, the filter will backwash that stuff out of the mineral, and if not, you'll be replacing the mineral (which should never be needed) instead of just adding to it periodically. And when you add mineral, you need to backwash the filter very good (maybe 20 minutes) before adding new mineral. To do that, you put it in backwash and unplug the control valve and time it. Then you shut off your water and add mineral and turn on the water and backwash the new mineral until the water going to drain is as clear as you can get it. So use a short piece (like a foot) of opaque 5/8" OD PE drain line so you can see the water leaving the filter. Otherwise you walk to the end of your drain line at the air gap.
If you do not give up the idea of that prefilter, you will regret that decision in a few months. And by the time you know there is a problem, it's too late for the mineral.
As to the drain line... you bought Vinyl tubing, do not use it. It kinks closed very easily.
The 5600 needs a 1/2" MPT x insert (barbed) fitting and you should go the foot of opaque PE drain line I mentioned above to 3/4" PVC. Then 3/4" PVC from the Clack 3/4" MPT drain line elbow fitting to the side of a 3/4" x 3/4" x 3/4" PVC Tee in the filter drain line. No check valves. So you don't need the insert x insert x insert Tee you bought.
I previously said to install this equipment on the left of your main water line as in the first picture you posted at the beginning of the thread.... you already have it on the left side so my error, that should be on the RIGHT side of the water main line. Water main coming up from the floor and the AN filter to its right and the softener to the right of the AN filter. Then back to the water main line.