The Logitech Z623 is one of the best choices for home DJing by far.
https://www.amazon.com/Logitech-Z623-Watt-Speaker-System/dp/B003VAHYTG
The JBL LSR 305's are a good choice if you plan on producing - but even if you do it's nice to have the Z623's still for DJing. The subwoofer is huge for DJing compared to JBL 305's without a sub. If you produce later on buy the JBL later and keep the Logitech Z623's for DJing. They are such great speakers and handy to have for anything, you could even have decent sized house parties with those speakers.
I think you are better off renting for the outdoor events and just going with a decent 2.1 PC speaker system for the house parties. I've been a hobbiest DJ for 15 years and done several < 20 people house parties on large enough PC speakers.
$500 is just not enough to buy anything worthwhile unless you hit the craigslist lottery.
I'd get Logitech Z623's for DJ practice at home and small house parties. They are loud enough for 12-20 people indoors and plenty loud enough for home practice. A 130W subwoofer on those deliver plenty of bass for the price. You could even get 2 of these systems and hook them up together. At $260 they would cost less and probably sound better than any $500 PA system out there. The $500 PA systems are truly garbage especially for DJing. They are fine for a small acoustic performance at a coffee shop or public speaking events but not where you need thumping bass of dance music.
Anytime you do an outdoor event rent and see if the party goers will kick in a few $$ for the sound system. The added benefit is that you'll get some experience with a few common PA brands. JBL EON series are pretty common to get from a rental shop and give you a pretty good introduction to mobile PA systems.
Hey man I checked out your pick, and i like it, but i did a little bit of searching and for the price of the bluetooth option on the one you picked, you can get this one. The reviews are better too. https://www.amazon.com/dp/B003VAHYTG?ref=emc_b_5_t Expect to hear more for me, i need to do some searching too so imma help you out bro. If you don't mind me asking, what is your price range?
You want amplied speakers with a subwoofer. You have no other audio outs on the TV that the 1/8"mm headphone jack?
"1/8" 2.1 powered speakers subwoofer
For instance http://smile.amazon.com/Logitech-980-000402-Speaker-System-Black/dp/B003VAHYTG/ref=pd_cp_pc_0