Can one get a head start on Shnayim Mikra? Would it count if someone either started early (perhaps during the weeks with no Sedra in Tishrei), or just had some spare time and went on to the next week's (and then the next and the next) Parashah?
According to Rabbi Israel Joseph haCohen Rappaport, the earliest time that you can start saying Shnayim Mikra B'Dieved is from Shabbos Mincha for the next week and L'Chatchila from Sunday.
I don't think it would work. The Gemara (TB Berakhoth 8a) says to finish it with the tzibur, not ahead of the tzibur.
Mishneh Torah, in Hilkhot Tefilah u-Bhirkat Kohanim 13:25, says:
אף על פי שאדם שומע כל התורה כולה, בכל שנתו בציבור, חייב לקרות לעצמו בכל שבוע ושבוע, סדר של אותה שבת--שניים מקרא, ואחד תרגום
In English( taken from here):
Although a person hears the entire Torah [portion] each Sabbath [when it is read] communally, he is obligated to study on his own each week the sidrah of that week, reading it twice in the original and once in the Aramaic translation.
In short, the answer is: No, one can't fulfill the obligation of Shenayim Miqra ve-Echad Targum for that week when starting before the end of the previous week's public Torah reading.
Yes! There is a statement on this webpage that seems to imply that certain poskim consider Shnayim Mikra to be an annual obligation, not a weekly one. Consequently, a person would be permitted to "get a head start on" upcoming Torah portions as long as he doesn't start before Simchat Torah.