Coups typically use the power of the existing government for its own takeover.

The Likud party, not Ariel Sharon, and certainly not Ehud Olmert won the elections. Certainly by default, the ministerial positions (prime or otherwise) should not have gone with them. But that is only the least of it.

Yesterday Peres, Itsik, and Ramon were required to quit both the government and as MKs so they could legally run in the next elections.

From Haaretz:

Article 6 of the Basic Law on The Government stipulates that MKs who quit one faction without resigning from the Knesset cannot serve as ministers as members of another faction during the current government term. On the other hand, once they quit the Knesset they cannot be appointed to the cabinet because by law, even in a transition government all ministers must be MKs.

Article 6A of the Basic Law on the Knesset bars an MK who leaves his or her faction, without resigning from office at the time, from running in the next Knesset election for a party that was represented in the outgoing Knesset.

Below is the actual current Basic Law text.

Basic Law: The Government:

6(e) A Knesset member seceding from his faction and failing to tender his resignation as a Knesset member may not be appointed as a Minister during the period of service of that Knesset. This does not apply to the splitting of a faction as defined by law. "Secession from a faction" is defined in section 6(a) of The Basic Law: The Knesset.

Basic Law: The Knesset:
The English translation above (from the Knesset) is slightly incorrect. It is not part of the main text, but rather in a Seif Katan 6(a) (sub-section) of another section of the law that is not posted on the Knesset website.

I haven't yet found an actual copy of the text, just references to it.

Anyway, according to the law, Sharon couldn't take the Likud name with him, because the majority of the Likud MKs stayed with Likud.

At this point, I'm no claim to understand what in the world is going on here.

But, it certainly seems to me it seems that there is something seriously wrong here.

The Likud still has 27 seats in the Knesset, Kadima has perhaps 14.
27 - 14 = 13 seats less.

Question: Why exactly was Sharon allowed to take the PM and ministerial positions with him, when he wasn’t even allowed to take the Likud name?

Answer: Technically, this has to do with laws regarding new elections, interim governments, and other unintended and completely unexpected applications of, and holes in, the law.

But in short, Ariel Sharon, via technicalities and gaps in the law, with a grand total of less than 12% of the Knesset seats, completely took over the government without requiring any majority support whatsoever, and now Ehud Olmert, replacing him, has free reign (I like that pun).

In short, Coup d’etat.