The "Smart Zombie" scenes:

There are five incidents like this in the series, and - not coincidentally - they all happen within the first two episodes.

Episode 1:

• The zombie girl picking up the bear

• Morgan's zombie wife turning a doorknob and looking through a peephole.

Episode 2:

• A zombie picking up a rock and using it to break a window

• A zombie starting to climb a ladder

• Zombies climbing a fence

Why does this happen? I see two explanations - one in-universe, one out-of-universe.

In-Universe:

Series creator and executive producer Robert Kirkman responded to this in a Reddit Q&A:

Q: In the beginning of the show we saw walkers do things like using a rock to help bash the doors in or turning a door knob, is there a reason we've stopped seeing them do that?

A: Older zombies are less together and capable of doing things like that. Fresher zombies, which there were more of in season one, are able to do more than older, more rotted zombies.

Out-of-Universe:

I think the real answer, although we'll never know for sure, is simpler. Season 1, and part of season 2, were produced by Frank Darabont. He was fired early in the second season, and since then, the zombies on the show have been far more consistent, and stick to the so-called "zombie rules" found in the TWD comics and Max Brooks' works.

More significantly, the first and second episodes were also written by Darabont. The first and second episodes were - perhaps not coincidentally - the only episodes he wrote alone.

Robert Kirkman's comment is plausible, until one considers the comic books on which the show is based. Kirkman had complete creative control over the comics, and if you read them, you'll notice that no zombies ever show this kind of humanity or intelligence. Not in the first issue, not in the most recent issue, not in any issues in between.

The only logical explanation for this difference, as I see it, is that Kirkman's answer to the Reddit question was an attempt to rationalize something that he would never have done himself, and that violates the rules his own work adheres to. In other words, Darabont didn't know that Kirkman zombies can't/won't do things like that. Someone probably corrected him after the first two episodes were filmed, because it never happened again, even while he was still the showrunner.

It is clear that Kirkman and Darabont butted heads, although we don't know if the issue of smart vs. stupid zombies was a factor:

There was a personal rift between Kirkman and Darabont...
- Darabont's replacement as showrunner, Glen Mazzara

However, Kirkman did a Nerdist podcast in which he mentioned his displeasure with the zombies climbing a fence in episode 2 - he specifically said he was unhappy with that shot, because if the show maintained consistency with that level of zombie mobility, it would cause enormous problems with the prison storyline.

There is a zombie that bugs me, if I could just point out a mistake, 'cause it's fun. In episode two of the Walking Dead series, there's a zombie that, like, totally scales a fence. And you're like "What are you DOING? They're gonna be in a prison in like, ten episodes, with a FENCE around it! Don't let that guy do THAT!"
- Robert Kirkman, Nerdist Podcast, Episode 93 (the quote occurs at about 1:17:00 in this video)

He also said something along these lines on the after-show, Talking Dead (I'm looking for direct quotes about this now).

However, he has been most vocal about his regret over the first season's finale episode set at the CDC, called "TS-19":

If I had to do it again, I wouldn't have done the CDC episode. It possibly gave away too much information and was such a big change very early on in the series. I feel like there might have been a better way to wrap up the first season. ...There were things in that episode that I think seem very much not of The Walking Dead world.

I probably would have changed that stuff. I've been careful in the comic series to not say what's happening in other parts of the world. It's something that's going to be fun to explore in the spinoff series. But the fact that France is mentioned in that episode and other things like that, I probably would have steered away from that stuff if I had to do it all over again.
- Robert Kirkman, interviewed by Hollywood Reporter

And although Kirkman has been careful not to criticize Darabont directly, he has said that he supported AMC's decision to fire Darabont:

It would be wrong for me to go into any details on the various changeovers because for the most part I was really only on the sidelines during the changes. Although, I will state for the record that I do agree with AMC's decisions in each case and strongly feel they were only acting with the shows best interests in mind.
- Robert Kirkman in an AMA

Darabont, on the other hand, has been far more outspoken - he has filed a lawsuit against AMC, and appears to refer to AMC and the producers of The Walking Dead (including Kirkman) as "sociopaths":

There’s a deep commitment and emotional investment that happens when you create something that is very near and dear to you, and when that is torn asunder by sociopaths who don’t give a shit about your feelings or the feelings of your cast and crew because they have their own reasons to screw everybody, that doesn’t feel good.
- Frank Darabont in Variety

Why didn't they cut those very brief shots of smart zombies from the first two episodes before they aired? I don't know. Perhaps he insisted on keeping them in; perhaps they thought those shots added something to the story; perhaps no one cared enough to edit them out.

But make no mistake: after the "smart zombie" scenes of the first two episodes, the writers and producers were careful to prevent similar incidents. Glen Mazzara, who was initially Darabont's lieutenant, then his replacement, and who only worked as showrunner for another season and a half before he too was fired, said:

I wrote a scene in which a zombie climbed a ladder and everybody told me I was an idiot because zombies can’t do that... I was, like, if I wrote it down it happens. And they’re, like, no, no, no. So we cut that.
- Glen Mazzara