What is the height of breeze/wind? How high does a breeze/wind blow? (If you are out in the open and can feel the breeze blowing, till how high is this breeze felt?)
And what are the factors that affect it?
 A: This is a question with a large answer, so I will do my best to summarise the key points and provide links (in-text) to help answer it.
Thinking about this question and doing some reading (to remind myself), the particular aspect (in my understanding) you are referring to are air masses, which according to this University of Illinois WW2010 website is defined as:

An air mass is a large body of air that has similar temperature and moisture properties throughout. The best source regions for air masses are large flat areas where air can be stagnant long enough to take on the characteristics of the surface below.

Further information is included in the slides of "Air masses and fronts".
Note, the homogeneity of temperature and moisture is true for the horizontal plane, in regards to your question - the vertical homogeneity related to the breezes that we feel at the surface is affected by many factors, explained in the chapter "Chapter 12: Atmospheric Boundary Layer"

The lowest atmosphere is far from being a simple system. Its physics include
  a diurnal component (typically convection during the day and stratification at
  night), complications due to complex terrain (surface elements such as buildings, forests, hills and mountains) and larger weather events (replacement of
  air masses by prevailing winds; clouds and precipitation).

The Atmospheric Boundary Layer occupies about the lower 1km of the atmosphere (it can vary depending on latitude) where the atmosphere has the most contact with and is influenced most by the surface, through mainly thermal and mechanical means, explained by the reference above:

The mechanical contact arises from the friction exerted by the wind against the ground
  surface; this friction causes the wind to be sheared and creates turbulence. In
  the absence of thermal processes, i.e. when the ABL is said to be neutral,
  we expect a logarithmic velocity profile.

A diagram of the diurnal and dynamic nature of the atmospheric boundary layer is shown below:

Source (and further information): "Atmospheric Boundary Layer Structure" (Piironen, 1996).
Further information about the above can be found in "Characteristics of air parcels and masses"
So, the breeze you feel may only extend metres to kilometres, depending on the factors discussed above.
