Sure, that is possible.
That is possible, too. Generally speaking, pump-probe is a technique, which is useful for a whole bunch of experiments. It is very versatile and can be applied to a lot of situations, so there is no general answer to your questions, so let me just explain some examples.
In many - but not in all - experiments the transmission of the probe beam is of interest. So imagine you have a sample containing some two-level emitters, maybe atoms or quantum dots or whatever. Now you shine a weak pump pulse (resonant with some interesting transition) on that sample and measure the transmission. As you will see, most of the incoming pulse will be absorbed and there won't be much transmission. Now you turn on a strong pump beam, which is also resonant with the transition, but hits the sample under some angle, so that you do not measure the transmission of this beam. If the pump beam is strong enough, you will excite some of the two-level systems to the upper state. Accordingly, the transmission of the probe beam will increase, too, because there re now fewer atoms to excite. So compared to the first case, the differential transmission will have increased. If you choose an even large intensity for the pump beam, you might even see stimulated emission caused by the probe beam and therefore the intensity of the probe pulse after the sample will be higher than the bare probe beam alone. Now the really interesting thing is the high time resolution you can achieve with this kind of experiment.
If you use a strong pump beam, you might imagine that the excitation of the atoms has some temporal shape. The number of the atoms in the upper state will rise quickly and afterwards show some exponential decay due to spontaneous emission in most cases. Now you send the probe beam over some delay line so that you can change the relative time between the impact of the pump and the probe pulse. Now you scan the delay by moving the delay line and therefore get data about how many atoms are in the excited state at a time t after the pump pulse has arrived. As light is traveling so fast, you can get a time resolution in the femtosecond range by moving the delay line, which is one of the main advantages of this techniques.
As another application, you can also check, how fast spin polarization of an ensemble of two-level atoms decays. If you use a linearly polarized probe pulse, the polarization will be rotated proportionally to the strength of the magnetic field inside the matrial in between (Faraday rotation). So you can use a circularly polarized pump pulse to introduce some polarization of the spins inside the material, which will then undergo a free induction decay. Now you can again vary the arrival time of the linearly polarized probe pulse and check how much the polarization of the probe beam was rotated inside the sample and therefore you get to know the time dependence of the magnetic field inside the sample with high time resolution.
These are just two possible applications. In principle pump-probe techniques can be used any time, you can initialize some state and are able to get some information about this state by checking the transmitted light. Of course the probe must be weak compared to the pump. Otherwise the probe already changes the state you initialized drastically. If you use lock-in-amplifiers, you are even able to see very small effects.