Good on you for asking questions before dropping loads of cash needlessly!
Any 12V horn will plug into the wiring on your bike. Any 12V horn will work. You may need to find one that fits in your frame bracket, and at the worst you may need to splice on the correct fittings for the electrics so it plugs right in. I always take off the stock horn right away and go to the auto parts store and buy a Fiamm "Low F" tone horn, it's very loud. You can put on a used motorcycle horn too, but don't pay more than $10, they're common and cheap.
There are hundreds of inexpensive aftermarket rear turn signals available. Don't get LED ones unless you are willing to alter the wiring, they'll flash funny unless you buy an extra load equalizer.
You can get direct turn signal replacements from a company called Emgo, and your local dealer should be able to order them for you. If it's more than $50, you can look at some aftermarket ones that will look different but work just the same.
Plug wires don't have to be exactly stock. It is quite common (and I have done this) to buy the boot/fitting and the wire seperately. The fitting has a kind of screw inside and you simply screw the new wire onto the fitting. Try removing a wire from the coil to see how it is made, many of them also screw on the same way. Try screwing a wire out of the coil, if it screws on out you can do this easy and cheap fix. The boots are $5 each and the wire is maybe $2.50/foot. Aain, a good dealer will pull ou the big parts catalog (Parts Unlimited and Tucker Rocky are good distributors for all these parts) and show you what to do. Taking in your coil and plug wires will help. Don't let them sell you expensive stock parts, there's just no need.
The batteries you're looking at probably range from the old-style ones you fill and check every month to a sealed glass matt. Any of these kinds are fine! The price difference reflects the amount of maintenance you mus perform to keep them good. You will only have to buy a Batteries Plus battery every season if you refuse to check the fluid level every month! Otherwise if you take care of the battery and keep it charged it will last for years.
Before you toss the battery you have, see if it takes a charge. If it fits in the bike, and takes a charge, then the battery is not your problem. You need to check the connections to find why the battery isn't getting the juice your bike is making, or if it's a short that is draining the battery as it sits.
You can also check the charging system but you'll need a shop manual for that, I'm not qualified to tell you how to do that.
I work in a little tiny bike shop and I'm telling you to try to buy local and support them, but sometimes they just suck and you need to look elsewhere. If you need some help understanding about the coils and plug wires or whatever, just PM me and I'll get you some links. Your problems with your Honda are almost certainly CHEAP AND EASY to fix! Don't let any dealership tell you hundreds of dollars to get it running, just walk away and look elsewhere.
Here, after you get the bike going good, this is an excellent idea for older bikes:
http://www.wgcarbs.com/index.php?option=com_content&task=view&id=12&Itemid=26