In regards to evidence for the 15th of Shaban
It is reported from Sayyiduna `Ali (Allah be pleased with him) that the Prophet (Allah bless him & give him peace) said: "Let all of you spend the night of mid-Sha`ban in worship (i.e. partly) and its day in fasting. Allah descends to the nearest heaven during this night, beginning with sunset, and says: Is there no one asking forgiveness that I may forgive them? Is there no one asking sustenance that I may grant them sustenance? Is there no one under trial that I may relieve them? Is there not such-and-such, is there not such-and-such, and so forth until until dawn rises. [Narrated by Ibn Majah with a weak chain.]
There are many other narrations from the Companions and early Muslims confirming this matter, as mentioned by Ibn Rajab al-Hanbali in his Lataif al-Ma`arif, and others.
There is general consensus that weak hadiths may be acted upon for virtuous acts, such as voluntary fasting and prayer, as long as the hadith is not excessively weak, returns to a general basis in the Shariah, and one is not convinced that the Prophet (Allah bless him & give him peace) specifically prescribed it.