Electrophilic attack of X+ on double bond
These kind of reactions are called Electrophilic addition reactions. These are usually given by alkenes and alkynes. In your example, the electrophile is $\ce{Cl+}$. The alkene will act as a source of π electrons and will donate it to the electrophile which is in need of electrons. So, alkene is called as nucleophile in this reaction.


  Is it the double bond attacking the X+ ion or the other way around?


It depends on how you look at it. We can say that alkene polarizes the $\ce{Cl-Cl}$ bond by pushing electrons into the anti-bonding molecular orbital. This weakens the bond and hence electrophile is getting generated. In the next step,the electrophile attacks the double bond [or double bond attacks the electrophile] and hence electrophile gets added across across the double bond

Why does the 3-membered cyclic transition state form?

The cyclic transition state is just a temporary relief for the electrophile that's just been added. The π molecular orbital donates electrons to the electrophile.

While the p-orbitals in the π-bond overlap with the electrophile, each p-orbital forms a σ-bond. 


  But would the same thing happen if I put any other electrophile?


Generally,electrophilic addition reactions take place through carbocation intermediate.
But this reaction can take place through cyclic 3-membered intermediate,if the electrophile has a lone pair.

Apart from $\ce{Cl+,Br+}$ even $\ce{Hg^2+}$ (when the reagent is $\ce{Hg(OAc)2/H3O+}$) and $\ce{NO+}$ (when the reagent is $\ce{NOCl}$) can form the 3-membered cyclic intermediate.