Best recommended bullet weights for a 1:9 twist 16 inch barrel in 5.56

A .40 caliber bullet is just a bit larger than a 10mm bullet -- 10.16mm. The caliber size is actually the measurement of the barrel's diameter in hundredths of an inch, so a .40 caliber bullet is 40/100 of an inch.

1:7 would be ideal,but your rifle may shoot this ammo very well.you will just have to try it out.bullet placement is critical with a small caliber,but,300 yards should be ok,if you put where it needs to be

The best way to generate that chart is by shooting over a chrono and inputting the data into an external ballistics calculator, such as this one by Hornady:

Ballistics Calculator

It takes some work to gather the data, but it's much more accurate than any generic chart can be. I've used cards generated by a couple of different online calculators and have gotten better than acceptable results out of them with both my match barrelled 5.56 AR carbine and 7.61x51 Savage 10FCP. Good enough to ring 6" steels at 300 yards and 500 yards, respectively.

There are many other web-based calculators that can do the job, just search for "external ballistics calculator" for more.

Shoot it with .38 special cartridges to practice trigger control and function drills (less recoil). Save your .357 for hunting, self protection, and familiarization fire. The weight of the bullet is not overly important, construction is. hollow points for self defense and hunting medium game. Rule of thumb? Light weight and fast for humans, heavy weight and slower for medium game.

Here is some good reading for you.

! click is 1 inch @ 100 yards, provided you have the correct scope for the rifle.
If you are sighting in a .22 LR 40 grain match velocity bullet for 100 yards, the mid range trajectory (50 Yards) is 6" higher. The bullet has a 6" arc half way to your target. If you are sighting in a .270 with 150 grain blunt tail bullets for 250 yards, the mid range trajectory is .4 higher. It only has 4/10" arc half way.
You most match the scope to the gun, and the 1 click= 1 inch rule will apply. Otherwise, it's lock it into a righd mount, fire it and adjust the scope accordingly. If you want to change the range on the scope, it will be guesswork.
I use a Bushnell Banner 3-9X40 with BDC. If you have BDC, make sure you have the correct distance ring in for the caliber.. It will make all the difference in the world.

The reason your shots are stringing is because the barrel is touching the stock probably at the end of the forearm. As the barrel heats up, the problem gets worse. Try sanding out the barrel channel and free-floating the barrel all the way to the recoil lug. Next, torque the action in the stock the recommended amount with an inch pound torque wrench. I torque my Remington 700 at 45 inch pounds for the front screw and 35 inch pounds for the rear screw. If you have a plastic trigger guard, 35 inch pounds will be too much for the light plastic trigger guard.