Know almost nothing about cars. :embarassed:
Never too late to educate myself, though.
Question 1: What exactly happens when E.30 engine is tuned to 300bhp (just the map). I know words like "boost", but in layman term, what does tuning mean? Turbo rotates faster or something? :embarassed:
In very basic terms, you are basically improving the overall efficiency of the engine. The latent or stored energy in the fuel is used more "fully", which creates more heat in the combustion chambers, which then creates more pressure in the combustion chambers, therefore applying a greater force on the piston.
In specific terms, this can be achieved in a large variety of ways. Specifically regarding turbos, they wont actually spin notably any faster, but the higher boost comes from altering the pressure at which the "pressure release" valve opens.
Question 2: When I drive and the engine is not fully warmed up yet (water is not up to 90 degrees), if I go above 3000rpm (in 2nd and 3rd gears) the engine sounds very coarse. I started noticing it after the map. WHen it warms up, it is OK again. Anyone noticed? What could it be?
Eeeekk. That is very bad news for an engine.

When the coolant temp is below its normal operating temperature, you should never exceed 3,000rpm, in any gear. And you should never use more than half throttle either. OK, if you have just pulled out of a blind junction, and you instantly notice a BFO truck belting rapidy up your chuff, then naturally give it some beans until a "safe situation" has resumed.

As Chungster reports in an earler reply, the crucial issue is actually the oil temp, and because the oil is "thicker" than water, it takes longer to heat up. However, being as the oil sump holds roughly half the volume as the engine coolant, the oil should roughly mirror the coolant. As a general rule of thumb, in the Golf, I will wait about another minute or two in the summer, and slightly longer in the winter, from when the coolant gauge hits the normal. On my RS4, though, it takes much longer, being as the sump holds a full 10 litres of Castrols finest - which explains why it has a readout of the oil temp.