I was in a VERY similar position that you are in now and I hope that you take the time to read this post. There are a ton of people out there that are /pf or /fi experts and with the stigma of people buying cars that are bigger than what you 'need' (especially it seems like in this sub) it's quite possible that this will get buried. There are already quite a bit of "I'm better than you because my situation is X" in here, which is absolutely ridiculous... I'll give you my story first, and then I'll address some of the specific questions that you asked about and that were brought up here.
I came back home from Qatar in 2014 at ~3 years and 6 months later I bought a house. I've had the same car since I got in (end of 2010), a 2003 g35 sedan I got from my brother (which I still have) for a few K personal loan. There were never any issues on it, but it was auto and strictly a DD type beater at this point. It was paid off a while ago, my Roth IRA was fully funded and I had maxed my deployed savings plan that was transferred to an independent investment account after my return. I got it in my head that I should finally seriously go look for my personal childhood dream car - a e36 m3 sedan. I wanted to potentially get into racing and have always loved motorsports in general. I've done research on the cars before I went to the desert, and while I was there. It took me months of arguing with myself about the money better spent, and how much it would translate to in 15-20+ years. I was convinced it was a dumb idea. The particular model I stumbled on was 16k (pretty high for the car, but a '98 with 50k miles and everything near flawless inside/out as well as a clean carfax). I talked myself out of it only to have my friends and family convince me otherwise. I flew to Indiana to pick it up on the 4th of July weekend and drove it home. I got a 72 month loan at 3% even though I could have paid cash for reasons I'm not going to argue about over the internet. I've had issues with the car since and in the last 1.5 years of ownership I put on about 2k miles on the car not counting the drive back home as I didn't drive it while fixing it (or waiting for the parts) and even still, would only drive it a couple times a week.
EVERY SINGLE DAY I walk out of my house and get into my DD I look at the m3 and smile. Every time I sit in it, no matter what the circumstances at home or at work, I feel happy. I have a grin on my face with every turn I take, every time I climb through the gears, and every second I hit the power band doing so. I come home and look at it, and I feel better about my day. At cars/coffee people can't stop talking to me about how clean the car looks. I'm going to Italy in June and I've had offers quite a bit higher than what I initially paid for it from people in the BMW scene here. I'm having my stepdad drive it to his garage and park it there for the next 4+ years because it's a car I want for the rest of my life. (Also, the appreciation on those cars are getting kind of crazy in the case that I do decide that I 'need' to sell it).
I could (and admittedly, occasionally do) sit here and look at my bank account and think "man, I could put in another ~$300 (loan+insurance) in there every month", but like I said, I still make my annual roth contributions, I still put in extra money into my investment accounts, and on top of that I also have 10% going into my TSP. FWIW My buddy just upgraded his e46 non-m bmw to an evo 8 mr and is absolutely in LOVE with it.
Now to address some of your questions -
Should you get a "fun" car? It depends. If you are absolutely in love with the car and want to get it for the right reasons, it will potentially make you much happier. If it's just a "well it'd be nice" thing, don't bother. If your biggest fallback is "well it'll help raise my credit" don't do it. I'm 27, but I never had a car loan and my credit score has hovered around 800 for the last couple years. Just using CC's will be enough for that purpose.
Should I wait until I'm "established"? Look - there is a fine line between "we might die tomorrow" and "we're going to live forever so we can't afford an extra 50$ date night / month". Ultimately it also depends. Are you able to have your emergency fund (if you have one)? Are you able to save a healthy amount of money towards retirement? Are you certain that other responsibilities down the line will make you less able to have the car? For example, my ex encouraged me to get the car. The girl that I have dated shortly after and am going to marry in a few months would be upset that I'm ordering car parts (let alone a 15k loan for the car itself), but bitching about spending an extra $100 on the ring/buying a wedding dress for over $200. (Like they say, priorities).
For all intents and purposes, you are established now. You've made staff, you're saving money, you're being responsible with your money. YOUR money. YOURS TO DO WHAT YOU WANT WITH IT. You don't know what will happen to you in a year, five or ten. 20k is essentially not a big deal (especially at 4.5% apr) on a car that is an enthusiast car - meaning it will likely keep value pretty well. Again, you have shown you are responsible and if you meet the "I truly TRULY wish this car was in my garage right now" criteria, I say don't wait.
Financing in the military? Same as outside of the military. 4.5% is a bit on the high side, but like you said, there are reasons for it. At the end of the day, it's not terrible and since you're already responsible with your finances, it's not a dumb a1c dorm decision.