With near certainty, I'll say the "14" is your rim's internal width (at the bead seat) in mm. Most say Sheldon's chart is a bit conservative overall, but I'd say 38 is at the upper limit for that rim.
Keep in mind: (1) a narrower rim will tend to result in a slightly narrower inflated tire width than the same tire on a wider rim, and (2) tires tend to be nominally wider than their actual width due to the attractiveness of a lower weight tire. Not always, but very often.
As you exceed the upper boundary, two issues can occur. One is decreased traction during turning due to less sidewall support. The other is possible damage to the rim. Under inflation is a factor in the former, while excess psi can cause the latter.
I rode 28mm tires for many miles on rims with an interior width of 14mm. Personally, 32mm would have bbeen my upper limit on those. I prefer wider rims and now ride 30-32mm tires on rims with 19mm width.
Googling "freedom ryder sport tires" showed me tires in the 600-800g+ range. Those are heavy, and that weight, in combination with knobs, will kill a lively ride on paved roads and trails. You don't need to get 25 or even 28mm tires. What you want first is a lighter tire with better rolling resistance. Panaracer and Vittoria both offer excellent choices for reasonable money. Check the T-serv and Randonneur models from each. You can get down to 350g/tire with these in a 700x32. Higher quality tires in 28-32mm are a real sweet spot IMO.
As for upgrades, address the contact points first: saddle, grips, and pedals/shoes. Then go from there as desired.
With very few exceptions, you upgrade the entire wheel, not just the hub. But better wheels are usually the most expensive upgrade. Unless you've got a deal on something forthcoming, just ride what you've got now.
Front derailleurs are very simple and have no ability to generate anything. If pushing the thumb button doesn't move the derailleur over with enough authority, it's likely a simple cable tension issue. You can DIY, but REI will tend to it on your new bike.