Required temperature difference between land and sea for sea breeze
So wind is caused by differences in atmospheric pressure which in turn is a result of temperature induced density differences. 

1. Required temperature diference:

A quick search brought up this paper:
Miller et al. 2003 SEA BREEZE: STRUCTURE, FORECASTING, AND IMPACTS https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2003RG000124

They calculate that a 10 degree difference between land and sea over 20km results in an accelaration of 7.2*10E-3 m/s. Now, this is an estimate. There are many more variables that will affect the complex fluid dynamics of the system.

2. Wind Intensity vs dT.

Absolutely if you increase the temperature gradient (difference) the wind speed will be higher.

from the paper they mention the sea breeze of Alcoa Bay, South Africa, where the strongest sea breeze occurs in areas with nearby dune fields:


  "During the daytime the sand dunes create a region of superheated air immediately overhead, setting up a very strong land-sea temperature difference."


3. Wind Intensity vs dT/dt

Now this is where it gets complicated. So thinking of this in a mathematical sense, the rate of increase in temperature is not going to matter that much because when calculating the wind speed at a given time what matters most is the absolute temperature difference between land and sea.
  For a temperature difference to occur the rate of temp change has to be fairly rapid as the system is always trying to reach equilibrium (i.e. remove the temp difference). So naturally if you have a parcel of air, it will reach a higher temperature if the heating is fast vs slow. If it reaches a higher temperature then it will result in a greater wind velocity. But if you wanted to make a more complex model of the system then you would need to look into the rate of change of change in Temp.