Time as an independent variable in multiple regression Can we chose time (t) as one of the independent variables in multiple (OLS) regression?
 A: In my opinion ,,, There is no violation in using time as an independent variable thus Yes "one can include time as an independent/predictor". Auto-regression versus linear regression of x(t)-with-t for modelling time series includes a discussion and a worked example.
The problem comes when one tries to use powers of time like time squared , time cubed etc as predictor variables for purposes of extrapolation (and even interpolation) . See How to extrapolate this simple trend line into the future for the purpose of forecasting in Matlab? @Ben's comments about the dangers of polynomial fitting.
Note that many "EARLIER" textbooks in the BC (BEFORE CORRELATION) era , used examples of fitting time polynomials to data .... seriously flawed in my opinion.
WARNING : BAD APPROACH FOLLOWS !
As an example of what not to do ( always dangerous ! )... Here is quarterly data from an "easy to read popular forecasting text"  . Yielding the following not-so-useful equation (user-specified)   with an incredibly deficient residual plot here  and Actual/Fit/Forecast here  . 
A: This is a very open question as you're not specifying what type of time measurement you want to add to your model, but the answer is always yes. 
If you're thinking about a linear relationship with time (e.g. age vs time) then you can always represent a date as a number of milliseconds or days and add it to your model as a regular continuous variable (your age is explained by the number of milliseconds since a date X, and 8.64e7 milliseconds extra will add 1/365 years to your age). 
On the other hand, if you're thinking about a not so linear effect of a date (seasonality, e.g. increase in sales in December due to Christmas), then you'd need to transform the part of the date you're interested in into a categorical variable (in this case, a binary feature that will tell you whether or not you're in December).
A: The answer depends on the underlying process. If the true underlying process is $y_i=\alpha+v t_i+\beta X_i +\varepsilon_i$ then yes, you can regress on time. For instance, you are measuring the temperature of an electric kettle when boiling water. Suppose, your $X$ variables are weight of a kettle, power rating of a kettle, amount of water, the temperature in the room. You're trying different kettles and filling with various amount of water etc. In other words, you're controlling the experiment. 
