I'm palying NWN 1 EE, and I want to make a melee fighter that can also sneak attack and maybe sneak up/scout stealthily, does anyone have any suggestions for a character like that?

How many levels of each class should I take? What should my stats look like? Is it possible to sneak in heavy armor? If not, should I go with medium?

Always start with rogue and not fighter. Since you get four times your normal amount of skill points for your starting class and rogue gets more skill points per level than fighter, you always start with rogue first.

I usually do Rogue 16 / Fighter 4.

Your remaining questions about stats and equipment depend entirely on what sort of character build you follow - are you going STR or DEX.

I usually use a DEX build, which is more about not getting hit than hitting things hard. Instead of heavy armor, I'm going to the wizard shop because DEX build gets better AC in flimsy robes than it does in heavy armor. (Fortunately, there is CCOHEE to help make my rogue not look like a wizard.) This does mean that I'm totally clueless about sneaking around in heavy armor. I'm sure it could be done, but I haven't tried it.

However there are situations, such as this one canceled premium module where I found myself in a solo arena battle against an enemy that I'd never be able to kill as a DEX build rogue as it regenerated hit points too quickly. It was missing a lot and not really hitting me either. Fortunately, nobody cleaned up between battles so there was a wand to summon a Dire Spider just waiting for me to pick it up as I do put skill points into Use Magic Device because it is fun to sneak up on a bunch of enemies and drop a Wail of the Banshee on them.

I want my guy to be mostly a strength-based swordfighter, but I'd like to be able to sneak attack and possibly sneak/hide too. Would having medium armor and/or a shield hamstring that?

I'm not the most knowledgeable guy when it comes to character building. But from what I know:

The heavier your armor and shield are, the higher your armor check penalty and the lower your max DEX bonus (in the case of armor). A low max DEX bonus would make no difference in the case of a STR rogue, as you wouldn't have enough DEX bonus to reach the cap anyway. But if you invest in STR instead of DEX, you will already be worse at stealthy stuff compared to a DEX rogue since you'd hardly get any DEX bonus to your skills, and armor and shield would additionally give you penalties, further reducing your chances of success. Combine that with multiclassing, which means you will have less skillpoints to invest in rogue abilities, and I think it's likely that most creatures that more or less match your level would spot you trying to sneak up on them.

Example for DEX bonus and penalties:
DEX 12 (+1)
Medium armor AC4, that is chain shirt or scale mail (-2) or Medium armor AC6, that is chainmail or breastplate (-5)
Small shield AC+1 (-1), or Large Shield AC+2 (-2), or Tower Shield AC+3 (-10)

So a fighter/rogue with DEX 12, lighter medium armor and a small shield would only get -2 to skill checks like sneaking (which might still work if your sneaking skills are high, but as multiclassing STR-based character they probably won't be), whereas a fighter/rogue with DEX12, stronger medium armor and large shield would get -6 already. If you try to sneak in full plate (-8) with tower shield, it would be -17 even, so a very low chance to stay undetected.

That being said, Knockdown would be a useful fighter feat for you to take, as knocked down enemies are considered "flat-footed" and as such vulnerable to sneak attacks even if you're not stealthy. And you also get sneak attacks on enemies that are distracted by fighting other targets, so any distractions, e.g. by henchmen, would work in your favor, too.

An alternative to multiclassing would be ranger as sneaky fighter class, because rangers can hide and move silently, too, but they are worse fighters and don't get the rogue's sneak attack.

Based on what Leroux stated, I think you may want to look into Fighter / Rogue / Weapon Master builds. However the rogue levels in these builds are not really there for sneak attack.

There are a ton of posts out there, both on the NW Vault and Reddit that will tell you that the only way to play rogue is Human Rogue/Fighter based on Strength. Neverwinter Nights is pretty forgiving as a game. You don't have to MiniMax your build to the extreme. It's okay to do some less optimal things, like how my rogue/fighter/SD gets an experience penalty for being an elf. Every race option brings something different to the table in terms of character builds. That being said....

I don't think anyone uses the Harper Scout prestige class as it doesn't add anything to any build that can use it. Likewise when anyone mentions Shadowdancer as part of their build, it is always one level of Shadowdancer for Hide in Plain Sight (HIPS) and nothing else.

I don't take Knockdown because I carry around wands to summon pets to tank for me. My other trick is to just lead enemies on a merry little chase after me, where I hit them then run away because you can sometimes catch them flat footed. Feydusters are also good for making enemies vulnerable to sneak attacks. Then there is Hide In Plain Sight that forces enemies to just forget that you're there until you attack them. If you're not into pets, running around, Feydusters or HIPS, then you really do want to invest into the Knockdown feat to make enemies vulnerable to sneak attacks.

Fighter/Rogue is my favorite combo. They work incredibly well together in NWN.

Strength is better than Dex in NWN, as you have more weapon options, don't need to waste a feat just to hit (Finesse) and will do more damage.

If really interested in stealth attacks and sneaking, I would suggest adding just one level of "Shadow Dancer" as your third class.

Sample build might look something like this.

http://world-of-greyhawk.github.io/builds/data/build204889.html