You said:

Allegedly there is a stone inside that Temple that was placed there by Mohammed himself

But, if you read the Wikipedia link you've posted, you'll see there's some misunderstanding involved in your question. From the Wiki page:

The Black Stone (Arabic: الحجر الأسود‎ al-Ḥajar al-Aswad) is the eastern cornerstone of the Kaaba…

…It was set intact into the Kaaba's wall by the Islamic prophet Muhammad in the year 605 A.D., five years before his first revelation. Since then it has been broken into a number of fragments and is now cemented into a silver frame in the side of the Kaaba.

That is to say, it's not a loose stone that just chills inside the Kaaba. It's a part of the structure itself. Is it possible that the fragments which were re-inserted into the structure got switched and aren't the original stone? I suppose since we don't have photographs of the process, anything's possible. But there's not much reason to doubt it's not the same stone, unless you know something I don't on the topic. Maybe the Qarmatians (scroll down) swapped it out, but I'd guess someone would have noticed, and there's not a reason to suspect such.

So while your question isn't really accurate to what's described in your own link, there's no reason to doubt that it's not the same stone (fragments now) as it was back in the 600s, without addressing how it actually got there since tradition is mostly what we have to go on.

from what I remember there are only small pieces of it left now.

It is embedded in one of the corners covered with a piece of silver that has a hole in it. It isn't in the center. Also Mohammed didn't put it there he only solved the argument Quraish's elders had about who puts it back in place

There's no reason to believe it isn't the same stone that was set there by Muhammad. In Islamic tradition and rituals, the Black Stone has huge importance. It marks the beginning and ending point of the ritual circumambulation performed during pilgrimages and traditions related to the Prophet recommend touching or kissing it if possible.

As such an important stone, there's no reason to believe that when the Ka'ba was rebuilt (as has happened numerous times in history), the Stone itself would be preserved and replaced in its original spot. Whether or not you believe that Muhammad placed it there himself, there's no reason to dispute its preservation since the early years of Islam.

There was a period in the 900s when Mecca was sacked by the heretic Qarmatians and the Stone was taken to Bahrain for about 20 years. It was returned upon payment of a ransom that the Qarmatians demanded. I suppose an argument could be made that the Qarmatians did not return the original stone, but there wouldn't be much of a reason for them to risk doing so, as they were a relatively weak and vulnerable state and antagonizing the Sunni world could prove to be suicidal.

There's absolutely no reason to think so - primarily because this story itself belongs to the genre of legend, not history. I'd say that this falls roughly within the same category of being a cultic relic; and the claims attached to these sorts of relics usually have no historical value at all.

There are similar things claimed elsewhere for Muhammad...like the Ahdnâme - the "charter" supposedly sent by Muḥammad to those of St. Catherine's Monastery in the Sinai (complete with his handprint!). Which is certainly a forgery.