I'm not sure what you mean by a dressage bridle, I don't think there is any such thing. The restrictions on what is allowed is more focused on the bit not the bridle.
Snaffle Bridle:
http://www.newrider.com/Starting_Out/Tack/bridle2.jpg
You can use a snaffle (regular) bridle at all but the highest levels. Most nosebands are also legal; flash, drop etc. but if you use a more unusual noseband check the rules. Most snaffle and pelham bits are allowed, but again check the rules if you use an unusual or harsher bit.
Double Bridle (dressage bridle?):
http://en.chevalpower.com/prodimages/LulworthWeymouth4084_large.jpg
You can use a double bridle at the higher levels and at the highest levels it's the only bridle you can use. This may be what you are calling a dressage bridle. It is used with a set of 2 bits, call the weymouth and bridoon, and used with 2 sets of reins. It doesn't look much different to the snaffle bridle, but has an extra strap to hold the extra bit. You can only use a cavesson or crack noseband if you are using a double bridle.
You can't use any type of bitless bridle.
The colour of the bridle should match the colour of the saddle.
I don't think there is any rule to say you must use a noseband, but most at least have a cavesson.
Unless you ride bitless, you current bridle is probably fine. Choose your tack to suit your horse, rather then the trends in the different disciplines.