Why does anybody, both in-universe and out think Pinkie is smart? I'll admit, I haven't seen every episode, but, to me at least, she comes off as downright mentally handicapped.

6441776
Couldn't have said it any better.

6441686
There are a few reasons that cause her to be found as 'smart'. And they are rather simple; found through observations, reasoning, and general attentiveness.

1) Eidetic (photographic) Memory:
She has shown that her memory is far exceeding average. She has memorized, not only the name of everyone who visits (or rather moves into) Ponyville, but the times and dates of their births (generally whichever she can find). And she has shown the ability to remember conversations (in the entirety).

2) Reasoning and Observational Skills:
She has been able to observe and reason things that most won't be able to just casually see and justify. The episode 'MMM-mm Mystery' is a stand-out situation as the writers had to put her in the extreme to show a lesson. But on several occasions she gave a statement that the others couldn't give... before acting in her 'odd behavior'.

3) Secret Location:
Her party cave is, for the most part, a complete secret from the town (and possibly the Cakes) until she showed it to her friends. That isn't something just anyone can do. Though, this point doesn't really help towards the intellect of her character, but rather her resources.

Really, it isn't very hard to come to that conclusion of intelligence. But, let's look at the opposite and reasons on why she is perceived as 'dumb' or 'handicapped'.

1) Odd Behavior:
She acts in a strange and almost idiotic way. Something that many ponies seem to balk at, and it has caused others to overlook her or outright ignore her. (To me, she is eccentric and puts on an act to be underestimated.) This is most likely brought on for laughs or for some other means.

2) Pinkie Sense:
(Okay, this isn't a point, but it needed to be addressed.) The 'Pinkie' Sense is hereditary, as Maud is shown to posses it and there is also the 'Cheese' Sense (was that what it was called?). So I can't say with any certainty that this is a trait of intellect, but rather something passed through her family.

3) Her Job:
Now, her job is to bring happiness (more or less). Let's look at the Jesters (Fools) of old, they were people who were hired to entertain courts; they would act like fools (hence the name) and bring happiness to those in the court -- but they were far from dumb, as they had to entertain in a 'smart' way. Pinkie will often act like a fool (this goes hand-in-hand with behavior) to entertain those around her, and while not everything 'sticks' they are attempts to bring happiness.

6441686

I think Pinkie is as smart as just about everyone else. I just think she really gets into her job, which is to make others happy.

She does have pretty good memory, now that I think about it, and she is very energetic.

6441686
Somepony who remembers all the dates of everybody's birthdays and has knowledge of getting rid of parasprites when no one else did can't not be smart.