What is the hybridization state of E (= Sn, C) in ECl₂?
Hybridisation is a mathematical concept, you never gain or lose energy due to it. Descriptions of the wave function involving hybrid orbitals or not are equally as valid. The linear combination of orbitals that result in the equilibrium geometry can however be interpreted as hybrid orbitals. Sometimes this may lead to a better understanding of the bonding situation.

From the above, it is obvious that hydbridisation is always a result from the molecular structure, never a cause for it. Hence we can deduce from the known bond angles the following:

Tin(II) chloride has a &angle;Cl-Sn-Cl of about 95&deg; (see Wikipedia). We know that p-orbitals are 90&deg; towards each other. Therefore the tin will have more or less unhybrididised orbitals. See also this previous answer of me and this answer of Ben Norris.
Dichlorocarbene on the other hand has a &angle;Cl-C-Cl of about 109&deg; from both experimental and calculated data (Phys. Chem. Chem. Phys. 2002, 4 (14), 3282–3288). We know that the angles between sp³ hybrid orbitals are tetrahedral angles of about 109.5&deg;. We can therefore deduce that the carbon employs sp³ hybrid orbitals in first order approximation.

The differences in the molecular structure and the utilisation of the s-orbitals for bonding stem from what is termed the inert pair effect, there is an explanation available in the this Q&amp;A: What is the inert pair effect?