Does a headstock with openings around the tuning pegs always mean it is a classical guitar or are there acoustic guitars with this design?

An acoustic guitar is one that's not electric, so yes, acoustic guitars can have slotted headstocks (which is what your picture shows). The wide rollers indicate that this particular acoustic guitar is a classical model intended for nylon strings. The wide rollers allow you to tune stretchy nylon strings without it taking forever.

Early steel-string guitars (like in my avatar) were an offshoot of the American gut-strung guitar, and also had slot heads. However the rollers were made of steel and were much narrower to allow precise tuning of the less elastic steel strings. Slot heads are still sometimes used on steel string guitars, especially those designed to evoke a "vintage" image, but solid headstocks are more common on modern steel-strings.

You also sometimes see slotted headstocks and classical style bridges on cheap Ebay guitars as Russell points out. These critters have narrow steel rollers and are sold (with minor design modifications such as the presence or absence of a pickguard) for use with either steel or nylon strings. If you run into one of these, you need to evaluate them on a case by case basis to determine what string type is most appropriate. Often the bridge looks like a classical type but is bolted down, so steel strings won't pull it off. The neck may or may not have steel reinforcement, and the internal bracing may or may not be strong enough to withstand the tension of steel strings. It can be tricky to make the call on some of these beasts. And since they're junk anyway, they tend to be barely playable without a lot of work and don't sound good no matter what you do!

A classical guitar is a type of acoustic guitar. Acoustic simply means that it is not electric. There are steel string acoustics and nylon string acoustics.

There are some designs that utilize a hybrid concept that has a steel string bridge (with bridge pins) and a slotted head similar to what you see on a nylon string guitar.

I have been working in music stores for decades and have seen some of these instruments before. In my experience they are very low quality and you should avoid them.

Acoustic guitars with a headstock like that are classical guitars meant for nylon strings.

Any guitar with a bridge like this is meant for nylon strings. Unfortunately, Ebay and amazon have cheap acoustic guitars that are classical guitars strung with steel strings that they sell for $15-30, paying no attention to the fact that this is wrong, but preying on the ignorant bargain hunting newbies.

Not recommended and only for short term as the neck will quickly warp because of string tension of steel strings.

You don't know much about guitars I guess!

All classical guitars are acoustic. If you mean, can steel string acoustic guitars have slotted headstocks (like the one in your picture)? Then the answer is yes, of course, although not usually.

This type of arrangement is called a "slot-head", and it's normally used on classical guitars as the larger-diameter tuning pegs are necessary to prevent slipping and account for the large diameter of the strings themselves.

However, slot-heads are often used for steel-string acoustics, especially older models or contemporary "parlor" guitars.

You can usually tell the difference as the guitars intended for steel strings will have a small-diameter peg of steel rather than the large, plastic peg.

And Tony is correct...Both are acoustic guitars.

check out the Martin d15s. theres more, but this will answer your question.