How soon is too soon to keg and drink?

Noob question here. Brewed a brown ale on sept 2. Recipe below. 1.050 OG, est. FG is 1.014. It will be 3 weeks in primary this coming weekend. I want to keg it, carb it for a week, and drink it. Ran out of homebrew last weekend. Bad, bad. does this need to age any longer? i'm new to kegging and am used to having to wait for bottle conditioning. does kegging eliminate that? Also, i'm limited on equipment right now but still want to keep a pipeline going...is it safe to keg and store at room temp (as long as i get the oxygen out)? if so, for how long?
Thanks for the help!

American Brown Ale – Extract
Recipe for 5.25 gallons (19.8 L)

7.25 lb pale malt extract syrup (3.29 kg)
8 oz 90˚ L crystal malt (227 g)
8 oz 60˚ L crystal malt (227 g)
6.5 oz chocolate malt (184 g)
1 oz black patent malt (28 g)
1 oz Willamette whole hops, 5.7% a.a. (21 g) (80 min)
0.75 oz Willamette whole hops, 5.7% a.a. (21g) (30 min)
1.25 oz Mt Hood whole hops, 6.1% a.a. (35 g) (15 min)
0.75 oz Willamette whole hops, 5.7% a.a. (21 g) (5 min)
1 tablet Whirlfloc or 0.75 tsp (3 g) Irish moss (15 min)
2 pkgs Safeale US-05

I'd say keg it! Force carb at 30 psi for 2 days and serve at 6psi day 3! That's what i usually do

Do you shake the keg up or anything or just leave it at 30 then turn it down

It's "too soon" if it doesn't taste good.

With that grain bill, I'd think you are likely approaching "drinkable" but you'll likely have a better beer in a few more weeks.

You can store beer at room temp as long as you purge the O2. Optimally, you would stick with cellar temps instead of 70+ if you want a longer aging. A few weeks to build up a pipeline isn't going to hurt though.

Thanks bill. I'm in Florida so mid to low 70s is my cellar temp.
So when you day a few more weeks and it'd be better, you mean in primary or leg and age?

Obviously I haven't tasted your beer, but most brews with that many different dark malts tend to keep improving for at least the first month or two. That will happen in bottle or kegs or secondaries or even the primary.

You should be good at three weeks as far as fermentation being complete. I would take a quick sample and see if it tastes good, (not ok) but "good" then keg it put it in your keezer or fridge and set to 30psi for about 36 hours and then purge the keg and set the regulator to about 10 psi. Drinkable beer in about 5 days from kegging date. So if you need it for a Saturday, keg on Monday night before the party.

Thanks for the advice. FG was 1.016. I kegged it Sunday. Did a taste test tonight and its not carved to where I want it but close. However the beer is pretty green. Co2 is on 12 and its at 38 degrees. By Saturday it should be where I want it for psi but will it age and mellow if I keep it this cold?

It will mellow.

It will taste better a few degrees warmer. You're not brewing Bud Light. A good ale will taste best and release more flavor closer to 50 degrees+.

This was one of my biggest mistakes when I got started brewing and I actually leave store bought bottles on the counter for a while now before I open. MUCH better results.

Ok. It has been 2 more weeks. I'm still disappointed in the lack of flavor. I turned up the PSI to 16 (I think my regulator is screwy). That helped after 3 or 4 days on the carbonation. However, I feel like there little flavor at all from the steeping grains. Isn't that where all the flavor in extract brewing comes from? I steeped for 20 minutes in 2.5 gal. I don't rinse the grains when removing. Could that be the issue? Is there any best practices out there to make sure the steeping grains are efficiently used?
Thanks!