Unfortunately you've named several purposes that somewhat contradict each other as far as barrel length, gauge, etc., so you would have to compromise a little. Assuming that home defense is your last priority for this gun (that would be my suggestion, as you can buy a second, much cheaper shotgun for home defense that truly fits that purpose), then a double in 12 or 20 gauge is your best option. I prefer a 20 gauge because they are a little trimmer and offer about the same capabilities as the 12 gauge--just my personal preference. I would recommend the following brands in your price range: Browning, Beretta, Remington, Winchester, or SKB. Browning and SKB probably offer the most options in your price range, especially if you want a 20 gauge. I personally like side by sides, but they are more expensive for the same quality, and unless you can find a nice used SKB 385 or 485 (still in the $2000 range) you will be around $3000 for a new Beretta, one of the few manufactures making a quality SxS under $3K. A good skeet or sporting clays version of any of the Manufacturer's OUs listed above would be your best choice--I would NOT recommend a Trap version--trap guns are built with higher combs and ribs to shoot higher than the point of aim as trap is a game of rising targets--they are not normally suited for hunting, skeet, or sporting clays. As far as barrel length, I think 30 inch is a great all-around length today--used to be 28 inch, and that is still good, but for skeet and sporting the longer barrels are becoming the norm (32 inch is becoming the favorite) but that is a little long for hunting IMHO. A 28 inch would be a great choice if hunting is #1, and would be OK for skeet and sporting. So pick them both up and see what you like--the longer barrel will swing smoother for skeet and sporting but also be slower in the field. I would look at a Browning Citori or Cynergy--I prefer the Cynergy because of its mechanical triggers, but they are both great and easy to find. SKB also if you have a dealer in your area. Beretta too, but usually a little more expensive. Can't go wrong with any of those. Good luck!
"Adversity does not make a man--it reveals him"