I also believe that based on your criteria, you have a couple caliber choices. I doubt you want to get into the NFA process at this point, so we'll stick to 16" overall length for both calibers - the 223/5.56 carbine that everyone is familiar with, or a 300 AAC Blackout carbine, that with the well-respected Barnes 115gr black tipped bullet, provides more kinetic energy and less penetration out of the same length of barrel.
I'm going to suggest to you that you build the lower buy yourself (easy to complete in 1 hour with youtube videos) and leave the upper to professionals. While this is a Build It Yourself forum, this is what I recommend for newcomers to the platform. Once you really feel comfortable with the platform, and you have the tools to do it correctly, by all means, build the upper.
Here's a good 5.56 build that I believe would serve your needs well.
Upper - PSA Cold Hammer Forged 16" with 13" Free Float Rail, A2 Flash Hider, and BCG/CH included. - $700
Sights - Magpul Gen 2 BUIS - $70
Total Upper Price: $770
Lower - PSA Blemished Lower - $40 (without transfer)
Lower Parts Kit - Red Barn Armory Build Your Own LPK -
$150
- Mission First Tactical G27 black grip
- RRA Ambi Star Fire/Safe Selector
- ALG ACT Trigger (owned by wife of Gisselle founder, great trigger feel with mil-spec pull and safety)
- Magpul MOE black trigger guard
- KNS Gen 2 Mod 2 Anti-Rotational Pins
- All other necessary pins, springs, and detents for lower
Stock: Magpul CTR Mil-Spec Stock Kit -
$75
Total Lower Price: $265 without transfer
Total Price: $1035
That's without shipping or transfer of the lower, so I'd add about $100 to that so you're looking at just over $1100 for a rifle with the following specs:
PSA Forged Upper and Lower
16" 5.56NATO CHF Barrel by FN with HPT/MPI tests, Chrome Lined Bore and Chamber, 1:7 twist w/ midlength gas system and A2 flash hider
SQH lightweight rifle length free float rail w/ keymod and QD slots
Carpenter 158 Steel Bolt and Full Auto Profle BCG
Misson First Tactical G27 grip
Magpul CTR stock and MOE trigger guard
RRA Ambi Safe/Fire Selector
ALG Advanced Combat Trigger (ACT)
KNS Anti-Rotational Pins
Magpul Gen 2 BUIS
To me, that's one hell of a rifle for around $1100. If I were to add a couple extra goodies, I'd add a BCM Mod 4 Charging Handle, and JP yellow springs to the trigger. Both easy and decently cheap upgrades to make the rifle absolutely awesome.
PSA has a good reputation here, look up some other builds using their "blemished" lowers and their uppers and you'll find it's hard to beat their price for the quality.
If you want to add a light, I'd add something like an Inforce WML that you can operate with your forward hand. I just checked the equipment exchange here and found decent red dots from companies like Primary Arms at $120 all the way up to Aimpoint at $500+
Hope something like this helps! You can always do some shopping around and beat the prices I've got listed.