The answer to the holding a weapon question is: sorta.
A cyborg in an anthroform body using a held rather than mounted weapon may use gunnery or the weapon skill, their choice.
A drone acting autonomously would use Targeting. Per the rules, the drone has to specify which targeting, such as Targeting (Longarms) or Targeting (Pistols). Or, more likely, Targeting (Gunnery). It's not clear what skill it needs to fire a held weapon - but that doesn't matter here, because you just pick that skill for its Targeting, whatever your GM rules it is.
Where it gets stick is what skill you roll when remote controlling or jumped in. There just isn't a rule, because you aren't a cyborg. So maybe you roll the weapon skill, maybe you roll gunnery, maybe you roll your choice. This may be a problem for you.
With regard to the drone's upgrades:
First, based on your requirements, the Lockheed Arachne is probably your best bet - the big thing it has going for it is +4 handling, which is really nice and adds to a lot of tests, including Infiltration.
As far as hardware/software goes, I suggest being pretty restrained about that. The Arachne starts out with a 4 Pilot, 5 Response, 5 Firewall, Clearsight 4, and Targeting 4. And it's only 1900ny, which is quite cheap. You could start upgrading its ratings...but you'd enormously increase the price of the drone for a comparably marginal effectiveness increase. I'd keep it economical and just get Covert Ops 3 and maybe Adaptability 3, depending how big your GM is on the Pilot tests to not be stupid (there's a huge price jump from 1500 to 4000 going from rating 3 to 4 - not worth 1 die, IMO). You do want better Sensors; rip its out, and put in something like this:
Sensor Package (2300) for Arachne
-Camera 4 400Y [1]
-Smartlink 500Y
-Thermo 100Y
-Vision Magnification 100Y
-Vision Enhancement 3 300Y
-Radio Signal Scanner 6 150Y[1]
-Motion Sensor - 50Y [1]
-Microphone 6 300Y [1]
-Spatial Recognizer 100Y
-Audio Enhancement 3 300Y
which gets it to Sensor 6, gets enough channels for a rating 4 tacnet, and gets Smartlink.
Then slave it to a commlink running a TacSoft, and make sure that commlink is well defended. Better yet, get more of them (at 5700ny each, plus the cost of a weapon to go into their mount, say 7000, they are a steal; you can buy a TacSoft and six of them for half your budget).
If you really don't want to do that, ditch most of the vision enhancement (I'd keep Camera 5, Microphone 6, Audio Enhancement/Vision Enhancement 3, and Smartlink, but lose everything else), put Analyze 3 on it, and keep it encrypted and running in hidden mode. You can't really get more than token matrix defense on it, though, not without spending enough money to defeat the whole point of using a simple, cheap drone.
I