Assume that no airline will allow visible tats any time soon. We'll have beards long before that.
Full sleeves are fine as long as you wear long sleeves that cover them. So you might need thicker shirts. You would not be the only one.
At every airline I've worked at or heard of, it's your choice to wear long or short sleeves.
As far as jacket, there's a couple problems with that. Doing walk-arounds in 100+ degree heat / humidity in the summer will be rough. Cockpits can be very hot before the APU is started, which is usually 10-15 prior to departure, so you'd be doing all the pre-flight stuff potentially in sweltering heat. Depends on the plane and airline though. Also since there's a "visible tat" policy, some captains at some old-skool airlines might have an issue with an open display in the cockpit... somewhat understandable if you have an FAA or company inspector on the jumpseat. I wouldn't plan on the jacket, just long sleeves.
The military is not an apples to apples to airline pilots. I'm guessing you won't be seeing any commissioned officers with full sleeves, unless maybe they came up through the ranks.