For general information about how Schengen days are counted, see How does the Schengen 90/180 rule work?
Is it necessary to go through passport control in Fiumicino airport? I'll be staying in that airport for 1:55 hours.
No. Most Schengen airports (every one I've been to) are constructed so that if you arrive from a non-Schengen destination and depart to a non-Schengen destination, you can transfer without going through passport control.
If I go through passport control, will the 90-days period start in Italy?
Yes, but only that day would count, because you're leaving the Schengen area on the same day. For example, if you spend 1 day in Italy and then 85 days in the UK, you could return to the Schengen area for up to 89 days.
For the sake of illustration, here is an analysis of your itinerary that assumes that you do go through passport control in Rome (for example, if you have time to have coffee with a friend in the airport lounge). I also assume some dates of travel to clarify the illustration:
Flight connection in Rome [on 11 April]
1 day in the Schengen area (if you go through exit controls after midnight, though, it would count as two days).
3 days in London (non-Schengen area) [arriving 11 April, departing 14 April]
This is only two days outside the Schengen area, because you will be reentering the Schengen area on 14 April. The only days spent entirely outside the area are 12 and 13 April. Note that for immigration purposes, however, it counts as four days in the UK.
2 days in Austria (will I enter the Schengen area through Italy or Austria?)
2 days in Czech Republic.
9 days in Hungary.
[Altogether, arriving in the Schengen area on April 14th, leaving on April 27th]
Schengen day counting is by the calendar day, or part thereof. So by arriving on the 14th and leaving on the 27th, you are spending 14 days in the Schengen area.
If you fly on Alitalia with a connection in Italy, you will clear Schengen entry controls in Italy. The flight between Italy and Austria will be an "internal" flight, which is essentially like a domestic flight. Your passport may be checked by officials at the arrival gate, or indeed anywhere else, but these checks are not done routinely, and your passport will not be stamped.
3 days in London, again [27-39 April].
This is not relevant to your Schengen calculation, since the 27th of April was already counted in the previous item. (But note that for immigration purposes it actually counts as four days in the UK.)
Given this itinerary, if you go through passport control in Italy both on your way to London and on your way home to Chile, you will spend sixteen days in the Schengen area. In reality, with slightly less than two hours for your first transfer, you probably will not go through passport control then. If you also remain in the "non-Schengen" zone of the airport when you return to Chile, then only the main visit of 14 days will count.