Why does my screen flash red and make an odd sound every time I fast travel from point A to point B?
I have no idea, I've never fasttraveled, but it does sound weird.
If I were to go looking for dragons, where might I find them?
After you start the main quest they will find you

You'll be randomly attacked in the wilderness. You can also encounter dragons near the Word Walls on the surface - you'll usually be pointed to them during the MQ.
Are there any secrets to obtaining gold relatively quickly?
You can chop and sell wood, smith, enchant and sell daggers or jewellery, craft and sell potions, pickpocket people - depends on your playstyle. Most of your gold will be obtained by looting, though.
I killed a group of orcs in Gloombound Mine because I sought, and obtained, ebony ore, but upon my return to Whiterun I was attacked by everyone and their mother and now hold a bounty of 12,000+ gold over my head and have killed a large amount of Whiteruns' populace. I don't unsheathe my weapons, and wait for a guard to talk to me, but instead just get attacked without any warning, and not only in Whiterun, but everywhere I go with the exception of Winterhold College. How do I get people to stop trying to kill me?
The orc weren't hostile, were they? You probably attacked them first and it counted as murder. Alternatively, they perceived your mining of ebony as theft and attacked you. Either way, it was murder. If the bounty is too high, you won't be offered the chance to yield.
You need to leave the hold and wait three days. After this the guards should stop attacking you on sight and you'll be able to pay the fine or go to jail and serve your time.
Aside from the residence in Whiterun, are there any other purchasable properties that you can find throughout the game?
You can have a house in every major hold capital: Whiterun, Riften, Markarth, Windhelm and Solitude. You'll have to do a couple of quests for the Jarls before you're able to purchase them, though.
How do you obtain Daedric/ebony arrows?
When you're on a higher level you'll start randomly finding them, you'll also be able to buy them in shops.
What is the fastest way to level up your magic-related skills?
Cast spells in combat

Illusion can be leveled up by spamming Muffle. You can also conjure a creature and kill it with Destruction spells. However, I believe it's best to level up naturally, without grinding.
Is there a way to kill giants without getting flung thousands of feet into the air?
Ahh... Archery, magic, and if going melee, high health and armor rating. Or a stealth kill, but it'll be a while before you get your Sneak high enough.
What is the best weapon you can actually obtain, and keep, in this game aside from smithable weapons with [Legendary] status?
I have no idea. Many weapons are scaled and their power depends on the level you obtain them on.
Can you join as many factions as you wish to join? If so, what are the factions you may join?
You can join as many factions as you want (with one exception). These would be: Companions, College of Winterhold, Thieves Guild, Dark Brotherhood, Bards College. Stormcloaks and Imperial Legion are mutually exclusive and you'll be forced to choose only one.