1) It is permitted to purchase and move in to a new house during the year of aveilus for a parent.
2) Yes one may hang a wet towel before Shabbos in an inconspicuous place, but not on Shabbos itself. If the towel was wet enough that one could have squeezed water out of it at the commencement of Shabbos it is muktza the entire Shabbos, otherwise it may be used.
3) It is not a problem since the sign is in a different domain because of the doorway.
4) All of them are permitted.
5) It is allowed because there is no intention to pull hairs and it is not an inevitable outcome to make it a psik raisha. However it is still not polite.
6) One should blot gently to avoid squeezing out the blood, but there is no problem of dyeing the paper because it becomes dirty and is not enhanced.