Are units in your own army intelligent enough to not attack one enemy unit each, and also to generally attack units that they are counters to? What I'm wondering is, is using the (A) hotkey wise when ordering an army to attack or is it better to make sure to issue commands more specifically? Can you trust the A.I. to make OK decisions when attacking a group consisting of several unit types?

I'm pretty sure they just attack the first unit they can reach first. 
You'll have to do it manually. "Micro" is the term they use for that.

For your first one, paritally. Units have smart AI in the sense that they do not overkill like in SC1, which is why you can't use a zealot to bait siege tanks. But they do will not focus fire. It's probably best for you to shift-attack que up multiple target and they'll focus each of them down one by one.
 
For the second one, I don't think so. They just go for the closest unit.

I literally just found out about shift-attacking. Haha.

shift-attacking? hmm .. new to me
also, I've noticed that units will stop attacking a building if another enemy unit appears in-range, but only for their auto-attack

shift+action creates a  queue for those units to do. So you could have a worker go build then shift+right click minerals and after he's done building he'll return to mining.

I have a similar question and didn't want to make a duplicate thread. 
How is the A.I. path finding? Is it easy to keep units stay in formation (heavy armor ahead of weaker ones)? 

I haven't played the original and was thinking of getting this, but before I launched up some Dawn of War 1 to get used RTSs again. Kept running into the issue that units going to a location and would take different paths and split up. Sometimes tanks would just get confused and block off a path, also some of the weaker high dps vehicles would just run ahead of the army and get destroyed before the rest of the forces arrive.

" I have a similar question and didn't want to make a duplicate thread.  How is the A.I. path finding? Is it easy to keep units stay in formation (heavy armor ahead of weaker ones)?  I haven't played the original and was thinking of getting this, but before I launched up some Dawn of War 1 to get used RTSs again. Kept running into the issue that units going to a location and would take different paths and split up. Sometimes tanks would just get confused and block off a path, also some of the weaker high dps vehicles would just run ahead of the army and get destroyed before the rest of the forces arrive. "

" I have a similar question and didn't want to make a duplicate thread.  How is the A.I. path finding? Is it easy to keep units stay in formation (heavy armor ahead of weaker ones)?  I haven't played the original and was thinking of getting this, but before I launched up some Dawn of War 1 to get used RTSs again. Kept running into the issue that units going to a location and would take different paths and split up. Sometimes tanks would just get confused and block off a path, also some of the weaker high dps vehicles would just run ahead of the army and get destroyed before the rest of the forces arrive. "

You aren't going to be using set formations like something out of Age of Empires if that's what you are asking. That said, the unit AI and pathing is fucking light years ahead of BW.  Though, in a way it can be a lot worse than broodwar. I think the ability to group and unlimited # of units into 1 control group really enables a lot of bad habits in players, day9 called it 1 control group syndrome or something. Regardless of how good the unit grouping and pathing is, good flanking and a good unit spread is still incredibly important. I can't tell you how many TvZs I've played where the zerg will send their entire army at me in one giant blob, instead of splitting up their army into 2-3 groups and flanking.

To answer your actual question, units with different movement speeds will not stay together. Although it's not really that big of a deal, you just need to be a little more attentive to you army. Keeping important units on a separate hotkey helps as well.

" @Crixaliz:   I'd say the best solution to that problem is to not issue orders for huge groups over a large distance.  If you wanna send your army to attack a base, give them short orders to move maybe a screen at a time.  This will take longer, but it will keep them together in a decent formation. "

They attack the first thing they find usually. If they find a building, it dies, however, with the A hotkey attack they will switch in some cases. 
 
Basically, if they are attacking a building they will target a unit instead. (Example: Telling Mutalisks to attack near a command center will have them kill all of the SCVs, then the nearest buildings when the SCVs die. )
 
The only other switch I have noticed is a Defend Theyself switch. For Example here: if you send an air unit over with an 'A' command and they run into a bunch of Roaches and Zerglings, they will start to mop them up, but once one Hydralisk shows up and attacks one of them, they will kill the hydra as it poses a threat to them. 
 
Therefore, from what I have noticed, this is the scale of thought that the Attack AI goes through from most important to least: 
1: An enemy unit that can attack me. (Defend self) Note: structures will jump up here if they can attack the units (IE: Cannons, Turrets, Crawlers)
2: Any enemy unit. (Good for killing overlords, drones, SCVs that repair buildings etc.) 
3: Any enemy structure. (Nothing else to kill? I'll just blow up this damn Supply Depot) 
4: Nothing. This is where your units continue to advance to the point you designated. 
 
One Final Note: Starcraft 2 has fixed the Supply Depot wall issue that the AI had in SC1 (at least last time I played it). What I mean by this is that, if there is a building in the way of a unit getting in range to attack a threatening enemy, the unit will then destroy the building in order to get to that enemy. In Starcraft 1 the units would kind of bounce back an forth, confused.