Well the second part is easier to answer:
Secondly, if the pH meter with automatic temperature compensation is calibrated with the buffers at temperature 25 degree C, then even if the sample is at any temperature like 20 degree C, the meter will actually display the pH of the sample at calibration temperature i.e. 25 degree C, is it right?
Yup. That's exactly what the temperature compensation is for.
I am facing a few problems when taking pH measurements.
When measuring pH of a sample like water samples with pH meter, the reading does not stabilize and goes on increasing. If the sensor probe or sample vial is shaken to homogenize the sample, the pH reading increases rapidly say from 7.5 to 7.8, and then continues to increase swiftly. What should I do to get an actual pH reading of the sample.
Any help in this regard will be highly appreciated.
Borek's webpage probably goes into much better detail, however, you have to realize that a pH electrode is only good for measuring the pH a buffer, or a dilute weak acid or base, or a very dilute strong acid or base (and the last one can get dubious rapidly.) Plain water simply doesn't have the conductivity to make a useful reading. You can check this yourself, your samples are likely not reproducible -- you won't get the same reading later in the day from the same sample.
The longer you have to leave the electrode in the water, the worse your reading, and the worse the next reading will be. You're depleting the electrolyte through the junction. This happens whenever you leave the electrode in solution too long. You should probably change the filling solution and soak it in storage solution.
Also, your results are odd to me. Reasonably pure water isn't typically pH 7.0, usually 6.6-6.8 because of dissolved CO2. When someone say water's pH is 7, I know they've only read about it, and have never done it.
The classical way to take a pH electrode reading of water is to a a neutral salt, pure NaCl is the cheapest best one, to give the water some conductivity. IIRC, something like 0.1 M NaCl should do the trick.