I will try to collect all the pieces from all comments and the question.
Note: Most information about this can be found in the camera's manual, which I tried to link accordingly. This is a rough explanation; some things are described in great detail in the manual, other things can be found on photo.stackexchange.
Am I not able to pick the shutter speed freely in P mode?
No, you are not. p. 56 in the manual offers the following information about P mode:
To obtain a good exposure of the subject, the camera sets the exposure (shutter speed and aperture) automatically.
In P mode, you can either change the exposure compensation (see below), the ISO, and/or the relation between shutter speed and aperture. So if you want to have a fixed ISO of 800, but switch from f/4 and 1/250" to something with a greater depth of field and/or more motion blur, give the main dial a push. 6 clicks and it should become ISO 800, f/8 and 1/60", which offers the same exposure value.
How do I change the shutter speed?
To set the shutter speed, use either Shutter-priority (Tv) mode (only setting the shutter speed) or Manual (M) mode (setting both aperture and shutter speed). Bulb (B) mode would also work, but you would have to hold the button as long as you want the exposure to be - not useful for anything below 1", and usually not really useful without a timer-controller remote shutter control.
Is metering and EV Scale related?
Yes, it is. The EV indicator (p. 18 for the LCD representation, p. 19 for the viewfinder one - and p. 78 for general information.)
The EV indicator shows either your dialed-in exposure compensation (in all modes except M and B) or the result of the camera's metering (in M mode).
Think of the EV indicator as temperature readout display / adjustment knob, while the metering is the thermostat itself. You can see how hot (light) it is via the display or you can adjust the temperature (exposure compensation) via the display, while the thermostat (metering) is then working with your values and trying to get the radiators (aperture, shutter, sensor) to achieve the desired values.
Therefore, when you set exposure compensation to 0, the camera will try a "neutral" approach: a grey card should come out perfectly grey. If your shots are overexposed (because most of the subject is black), you can dial in a negative exposure compensation (left side of the scale), while for underexposure (because much of your subject is white) needs a positive compensation (right side of the scale).
In M mode, you have to do your own exposure - and the EV indicator helps you at that: it shows you how the camera believes your shot will be exposed. This is not as good information as a histogram (or, even better: a waveform), but it offers a bit of orientation.
Should shutter speed increase like this because of the change to spot metering?
Metering (p. 77), on the other hand, defines how the camera evaluates the frame. Spot metering, for example, will just evaluate the middle of the frame and nothing else. So if the middle of your frame is pitch black, but the surrounding area is white, then your exposure will be quite bright. As a rule of thumb, spot metering is best left alone unless you really know what you do.