No worries, cutting is fine!
At the University I went to for Equine science, every horse's mane was cut, and that's about 300 of them.
You just have to do it right.
After cutting it the length you want, take the scissors and cut up, into the hair, your scissor blades parallel to the hair shafts. You only cut the tips, and this makes them look a little tiny uneven, like if you pulled it.
Trick of the trade, right there. A lot of the show horses you see in the paint or quarter horse circuit will have a mane done this way
If your horse's mane is too thick... that's when you should pull it :)
For sun-bleaching (I have a bay too!) Try quic black on the mane and tail (helps hide the red-tones) then use a sheet on your horse (a full cover fly sheet works wonders). For the tail, condition it really really well, then put it up in a tail bag without washing it out (if theres a lot of bugs, tie some cloth to the tail bag so he has something to hit the bugs with). Not only will it keep the tail from drying out and bleaching, but it will make it grow nicely, too :)
If you're not that serious... then you could just turn your horse out at night if he has a stall, or... deal with bleached hair.
Sweat speeds up the bleaching process also, so after you're done working him, hose him off!