Can one get a haircut in the week before Shavuot (if so, starting what day?), or anytime in the period between Lag B’Omer and Shavuot? Can one bring an item to the tailor during the week before Shavuot to be worn on Shavuot?
Regarding taking haircuts during sefira, it depends what your custom is, as there are numerous customs, the basic idea of all of them is not to take a haircut for 33 (or 34) days. If you keep the first half, then you may take a haircut from the morning of Lag Baomer, (for Ashkenazim, and for Sefardim from the next morning of the 34th day). If you keep the second half of sefira, then it will depend. If you started from after Rosh Chodosh Iyar, then you may not take a haircut until Erev Shavuos (except for on Lag Baomer). However if you started before Rosh Chodesh, then you may take a haircut from the morning of the 3rd of Sivan, (which is the morning of the first of the shloshesh yimei hagbalah).
Regarding bringing clothes to a tailor is permitted unless you have a specific minhag not to buy or sew clothing during sefira.
O:CH 493-1,3, M:B ibid15, Biur Halacha D:H Yesh Nohagim, Igros Moshe O:CH 1-159, Piskei Teshuvos 493-3..