How big does a lake have to be to have its own Sea Breeze?
Lake breezes(similar to sea breezes) are fundamentally a feature of mesoscale meteorology and the peer reviewed reference Small Lake Daytime Breezes:
Some Observational and
Conceptual Evaluations details both the observational studies of lake breezes and the conceptual understanding behind the formation of the lake breeze.

Since OP's question is 


  How big does a body of water need to have a sea breeze?


The answer according to this book Climate in a small area which is one of the references in the first reference is that lake breezes have been observed in lakes with a width of about 4 kms(Lake Suwa, Japan ) and a width of 10 kms (Lake Constance, Switzerland). The lake breeze reached a speed of 2 m $s^{-1}$ in the case of Lake Suwa and slightly greater speed in the case of Lake Constance. But because both of these lakes are orographic lakes the presence of thermally induced upslope flows (diurnal heating and consequent upslope flows during the day)  can interfere with the formation of lake breeze. 

Further observational studies carried much later for the following lakes  Lake El Dorado , Lake Okeechobee, Lake Winnipeg and Lake Eufaula have reported lake breeze speed values in the range of 2-4 m $s^{-1}$. One must remember that these lakes are much larger in size but in essence the numbers capture the range of the lake breeze values. These observations had been obtained as a result of measurements carried out over a period of few months and over different weather conditions. 

The upper range for the lake breeze is around 6 m $s^{-1}$ and any lake having a width of less than 2 km is likely to have an undetectable lake breeze. 

So the physics behind the lake breeze formation in essence is similar to the physics behind the sea breeze formation. A pressure gradient(density gradient) is created between the land and the lake and this gradient is primarily established in the lower troposphere. Greater the inland sensible heat flux greater the value of the pressure gradient. 

The following URL shows satellite images of different stages of formation of a lake breeze during the day - Lake Breeze Satellite Imagery

In regards to the second question I believe steam devils are mesoscale vortices that have been observed to form over lakes. These vortices are no more than 50-200 meters wide and can develop in the absence of any major synoptic scale forcing. But the surface area of the lakes have to be in the range of Great Lakes of North America  in order for the strong surface heat fluxes to play a role in mesoscale vortex development. The following peer reviewed reference provides observational and modeling studies of steam devils- Mesoscale spiral vortex embedded within a lake