Is there a way to stop things from attacking my horse/NPCs and to pull them to me? Or Stop my horse/NPCs running off into combat?

If you have a high enough Illusion skill, you can use the Calm or Fear spells or their higher-level versions to prevent your follower or horse from attacking.

Mods or Console Commands

The mod Better Horses, can set horses to flee at the sight of combat, rather than getting involved and killed. This can be set to all horses except Frost and Shadowmere, or to all horses, including the mentioned two.

The Better Horses mod can also set your horse as 'essential' (i.e. unkillable) or increase your horse's health, preventing or lessening the chances of your horse getting killed. It also has other useful, optional horse-related features.

The mod, Better Stealth AI for Followers, claims to prevent followers from rushing into combat:

This mod improves the stealth AI for followers, so they don't rush into combat before the enemy has fully detected the player. Followers will also unequip torches while sneaking.

The mod Better Followers or Ultimate Follower Overhaul will allow you to set your NPC follower as essential, preventing them from getting killed.

Note that Better Followers and Ultimate Follower Overhaul will work for Permanent followers only and Quest followers may not be affected. Most quest followers are already set to be 'essential' though. Just be careful with those not marked as essential (see who is not set to be 'essential' here).

You could also use the setessential console command to set your horse or follower as unkillable.

I have looked everywhere and I can't find a way to keep our horses from attacking mobs in skyrim. The best thing I can figure out is to simply be aware of your surroundings, if you see something coming up that will attack your horse, dismount before getting there. I've even had to resort to running back on the horse to "park" the horse behind a rock or something, then dismount and attack my pursuers.

It's a pain to have to do some of that, but it does add to the realism of the game. If your horse dies you have to buy, steal, or find another.

As far as companions, you can talk to your companions and tell them to "wait here". Aside from that, they are free thinkers.

There is no "real" option for that, but to make sure check your companion's dialogue options. I personally didn't find that option in skyrim, but it existed in Fallout 3 or NV. Also, the community usually makes mods for things like that. It will take some time, however. Until then, you are better with as few followers as possible.

Also since you can use potions and heal yourself you should rush in the enemies or conjure something near them to attract them before your horse does that.

A few other things you can do to increase their chances of survivial depending on race/character/progress:

• As mage you can heal them or cast other protective spells, shields, etc...
• Battle cry or something like that. I believe every class has it (makes foes flee).
• Taunt: One race or class had an option to make close enemies focus on you for a period of time. I am not sure which one.
• Shouts: you get a shout to knock back enemies and the one to freeze them pretty early in the game.

If playing on PC I find the console commands useful. I open the command console, click on Shadowmere so his code comes up and then type kill and press enter. This will kill Shadowmere, but when you're done fighting simply open the console again, select Shadowmere, and use resurrect to bring him back to life. It's a useful strategy if you're willing to use the console.