why are you using a 1 shot railgun?

It's a good beginner design, and I don't see any critical flaws in it which will render it inoperable. That said, there are a few ways you can improve on it.

From top to bottom:

1. The DCR tells me you're using damage control, which weighs down the ship by good 150 tons without providing much benefit (lack of armor and shields will ensure it dies before making use of its damage control, and instant repairs will quickly run into lack of MSP). This weight is better used in more engineering, fuel, and deployment time.
1. 1. On topic of engineering, I'd recommend having at least twice as much MSP as Max Repair, to allow for on-board repairs of the most expensive part of your ship. Max Repair lists the cost of maintaining such part; repairs will require double.
2. Cryo berths also weigh it down a little, but I presume they're there for RP reasons. If they aren't, they can safely be removed; since the boat will be patrolling between populated colonies, there'll always be one close by to drop survivors and POWs.
3. The deployment time could be a little higher.
4. This ship is slow by the Quantum Singularity standards. If it's meant to move around escorting commercial ships, it's fine, and low-boost engines also makes the patrol ship very cheap; but keep in mind that at this tech level it's easy to push 50kkm/s or more.
5. The only thing in terms of defense this ship has is its ECM suite and range of its missiles. 5 points of shield is not stopping anything, and even if they did, you're better off making a properly-sized shield. The capacity of small shields is very heavily penalized.
5. 1. Not to mention that full ECM suite is good 450 tons of extra weight for your ship. Fire Control and Missile jammers can be very useful, but the Sensor one is a lot more dubious.
6. The railgun doesn't need such high capacitor recharge rate. 10cm railguns achieve their maximum rate of fire at Capacitor Recharge Rate 3, and you have 25. (The "Power" row in the description lists the power you need to fire the gun as first number, and the amount of power it's getting in 5 seconds as a second number. Excess power is wasted.)
7. Fire Control looks good, it just won't benefit much from electronic hardening. That only allows it to resist damage from microwaves, while driving the cost up by, i think, 2 times as much. Same applies to your MFCs and passive sensors.
7. 1. You will need an active sensor lock to target enemy ships, either from this ship or from an accompanying scout.
8. Missiles look pretty good to me, no issue here.

It was a long-standing debt to me to create visually matching flags for the race portraits, spacecraft and space station icons I had previously made. These have been completed today: a total of 600 custom flags with crests/logos. This finally completes my graphic package.
To install, simply copy them to the Aurora C\Flags folder!

Could the smart drag in the ground units OOB be made to have a non smart mode? Maybe a button to switch between smart drag and a specific function. Right now you cannot attach a smaller bombardment formation to support a larger formation with an HQ which is a common complaint and this would be a fairly simple resolution.

...
The population of Mars still shows as "0".

Antonin, in the Summary tab of the Economics window, the minimum Population value shown is 10,000.
Even if you see a "0", the people are there, simply they are too few.
So, unfortunately, it takes at least 25 travels of your colony ship to see some population (less, if people start to reproduce; more, if they start starving and dying).

Suggestion: you can ask the civilian lines to bring structures (infrastructure at first) where you want/need, to prepare or expand your colonies.
In the Civilian/Flags tab, add a demand where you want the structure(s), and add a supply where you produce them (or where you have a surplus).