what kind of home audio system should I get?

Cube speakers IE Bose forget that trash.

Your older speaker blow because you over powered them too much, so you will need something that is rated higher watts & look for continus watts not max watts.
Yes all blurays play CD's, but getting one depend on what sized HDTV you have more then anything else.

If you want something that will last you years and years, your going to need more then $1k, closed to $3k+ for a reciever and a 5.1 or 7.1 speaker setup. Now you do not have to get everything at one time, pick up a reciever for around $500-800+ and a center channel for $300ish.... Save up some more and get some fronts at around $200-400 each.... repeat... save then get a sub look at spending around $400+ on a good sub... repeat surrounds speakers and last back speakers. It may take you a year or two but it will last 10-15 years or longer of quity sound. And always keep with the same brand, the sub can be a diffrent brand as long as the crossover is in the range of the other main speakers.

I would not buy any used electronic off eBay.
HTIB systems always come with super cheap speakers and would not work in that sized room, that includes Bose. You would have to crack them at or near there limits and this reduces there life spand and can clip the speakers.

For receivers like Yamaha and Denon

I use JBL Studio speakers but there are many good brand out there, keep away from Sony, Bose, lowend JBL's models and Yamaha speakers. Mostly take your fav movie and CD into a 'good' electronic store have them loop it and play with diffrent speakers. And note how much power the reciever is pushing, make sure the speaker do not clip. A good setup you should not have to push the reciever pass 50%.... You should not have to crank any reciever over 80% just to 'hear' the speakers, if so forget those speakers, your just going to blow those out too. Rember continus watts not peak (or max watts), continus is what the speaker can handle with out damage, peak watts (what stores like BB will use too push sales to make them look good on paper) is just for a few seconds before damage can happen.
Like if you go into BB they will say 200wts so it sounds like 'lots', but really the continus watts is just 50-80wts. You see this on HTIB where it will say 1000watts, but what they do is include the powered Sub watts, a ' normal' setup the sub watts is not included. Also HTIB lie about the real watts all the time just to sell there junk, a '1000wt' system will really be like 200wts and you would have to push the speakers for a room your size.

1. Can I use just an amplifier since I don't need radio?
Yes you can, but you will find more bang for the buck just getting a receiver even if you never use the radio, the separate component world has moved more to the true audiophile and the stuff is usually expensive.

The other thing to consider is you will want as many audio decoders as you can get in your budget which should be just abour or all of them depending on your other requirements, power, etc...

Take a look at the Sony line up, I have had real good luck with their ES series, I have been real happy with the price performance. Look at something like the STR-DA2400ES. List price it looks like it will take most of your budget but you can easily get them new for 200 bucks less if you shop around. Very flexible and has a LOT of audio decoders.

2. Should I get rid of the dvd player and get a BlueRay?
I would not get rid of your DVD unless you don't have room to add a Blu-Ray to your mix and have them both, Blu-Ray players are very slow in things like loading the disc and changing menu's, if you have an HD TV and your DVD upconverts to at least 1080i I would consider keeping it, I know I kept mine and still use it.

That is a personal call, I would but I would not invest in top of the line at this time, they are still sort of shaking things out, if you can find the discontinued but still supported Panasonic BD-35 you will get a good Blu-Ray at a great price. Don't get a used one but you can find new ones on Ebay and Panasonic still honors the warranty and if you are real worried you can pick up like a SquareTrade warranty for a pretty low price.

If you do not have at least a 50" HDTV I would not bother with Blu-Ray at all. (Again my opinion...)

2a. If I get a BlueRay will that play music cds too?
Most, at least the BD-35 or BD-80K if you decide to spend more will play everything. As far as I know every BR player I have seen plays everything.

3. How many wall mounted cube speakers do I need for a 20x15 room?
5.1 or 7.1 the latest is the only way to go in my opinion, you can easily go 5.1 for now and add the other 2 later. At 5.1 you will will get a true surround sound, the other two just add some depth. Your receiver during config will ask how many speakers you have and deal with everything from there.

Unless you are really stuck on wall mounted, I would skip that part provided you can put the 5 or 7 speakers where they need to be for optimum surround, if not, then the wall it is.

You want Front L/R, Center, Rear L/R, and a powered Sub-woofer. If you have to mix or decide to mix, the 3 you want to get the best your budget fits are the front l/r and center, followed by a sub-woofer, then the l/r rear, and if you go 7.1 then the surrounds can be the cheapest. The sub-woofer is really essential to get the most out of your system and allows you to get a lot more kick with less power from your main amp/receiver.

As for the power, remember if you go at least 5.1 then the power per channel can be a lower than you old stereo rig, the subs are powered so that just just a line out from your receiver and you have twice as many speakers, you can rock the house pretty good on 200W/Ch on 5/7.1

On a personal note, I would skip the Bose, they are in my opinion an acquired taste, some swear by them and some hate them, I generally like JBL's and you can get complete JBL surround sets, all matched to play together well.

What ever you do make sure to get a center that is designed to be a center, this is where a lot of the content is coming from and if you have a cheap center, the rest no matter how good will not make up for it.

3. What kind of amp/receiver should I get if I want to run speakers into another living room across the house and possibly outdoors at a later date?

Any system that is set up with A, B, A+B/Both will do, that is, they have at least two set of speaker connections on the back and a way to pick set A, B, or both at the same time, any that have both will have these options. Again I like Sony. You can also get wireless external speakers, some receivers are set for this some are not, so you will want to decide how you plan to get the signal to your other set, wireless have powered speakers at the other end and will want a line out level usually, although you can get wireless that will take speaker out power and deal with it properly, I would go with getting just a receiver that has two set of speaker connections, that leaves your future options open.

4. Suggested brands for all equipment?
For me:
HDTV: Panasonic Plasma
Receiver: Sony
Blue-Ray DVD: Panasonic
Speakers: JBL

5. Is is a bad idea to buy a boxed set?
On a budget and for the least configuration and set-up a "boxed" Home Theater system takes a lot of the pain out of things and you tend to get more for your dollar, they typically include the Receiver, Blu-Ray Player, and Speaker Set. Will you have the best, no, will it be pretty good, all depends on what you spend, but for less money you are likely to do at least as well if not better than trying to get the best of each component. The other plus is one remote control and not 3 or more.

6. Is it a bad idea to buy used equip. on ebay?
For this kind of buy I would never suggest used from eBay, you never know why the person is really selling it, did they upgrade or is there an intermittant problem, or a solid problem, can you transfer the warrenty... If you are going to drop a grand you want to make sure you can get back to the seller, which means either retail or Ebay new and take the warranty path if there are problems.

Finally and again I will qualify this with in my opinion, not counting the HDTV for a grand you should be able to get a pretty good home theater, one that will rattle the windows if you want, just not top of the line and if you are not an audiophile as you mentioned, you should do ok on that budget.

Remember that the end product is delivered by the speakers, so that is where you do not want to go too cheap, they make the most difference these days. The specs on receivers within similar price ranges are all about the same these days the biggest difference on receivers is the flexibility, inputs/outputs, audio decoders, up-converters and up-scalers. Remember you are dealing with Composite or S-Video, Component, HDMI, and DVI video, then L/R analog, and Optical and Coax Digital audio, and you want to make sure you are able to hook up what you buy and do the things you want.

It is this complexity that just drives a lot of people to a boxed set, but like all things in life there are trade-offs, in the boxed set, it will all play together well, but may not play well with others, and on the flip side, getting one of everything from different companies puts it on you to see that they match up right.

Good Luck.