No, the pitcher will not get charged with the run, but since the other team scored a least 1 run it is no longer a shutout regardless how it scored.
A shutout occurs when one pitcher pitches a complete game without allowing any runs. If another pitcher comes in, neither is awarded a shutout, but the team can be said to have "shutout" the opposing team.
No, only if he completes the game does he get credit for a complete game. He also gets credit for a shutout only if he completes the game without allowing a run, if other pitchers complete the game without allowing any runs, it is a shared shutout which is not an official statistic.
In 1944, Charley "Red" Barrett of the Boston Braves threw only FIFTY-EIGHT pitches during a nine inning complete game. Barrett's Braves shutout the Reds 2 - 0 and the game set major league records for least number of pitches known to have been thrown by a single pitcher in a complete game and shortest game played at night.
Five innings is the minimum for a complete game, so the answer is yes.
Shutout
No--allowing one baserunner for any reason (including walks) immediately ends the perfect game. You may be thinking of a shutout (a game without any runs scored). Again, the answer would be no; in the situation you describe the run would be unearned, but would still prevent the pitcher from getting credit for a shutout.
I think it is Satchel Paige.
Pitch more shutout innings - that is, innings where you don't give up an earned run. ERA is calculated by taking the number of earned runs a pitcher gives up, dividing it by the number of innings he/she pitched, then multiplying the result by nine. So, for example, if a pitcher has six complete innings and gives up two earned runs, their ERA becomes three (2 divided by 6 is 1/3, 1/3 multiplied by nine is 3). This works over the course of a pitcher's career, so if a pitcher gives up six earned runs over six innings in one game, his/her ERA becomes nine. If he/she then pitches a complete game shutout (nine full innings, no earned runs) their ERA drops to 3.6 (as it is now six earned runs from fifteen innings).
Pedro Ramos ... Ramos threw a complete game 3 hit shutout as the Twins won 6-0.
Santana...shut out the Red Sox...
According to Baseball Reference, Maddux and the Braves defeated Dwight Gooden and the Yankees at Yankee Stadium, 2-0, on July 2, 1997. Maddux threw a three hit, complete game shutout with eight strikeouts and no walks. He threw 84 pitches and 61 of them were strikes.
Jordan Craft of the NCAA D-1 Dallas Baptist Patriots, threw a complete game shut out against Grand Prairie in 2006. Craft threw a complete game shut out, despite giving up 11 walks. There were 7 double plays turned in support by DBU.