My birthday is Asara B’Teves. Would it be proper to have music and dancing on the night of Asara B’Teves, or should I make the seudah on another night which is not the Bo Bayom? In otherwords, this year, Asara B’Teves comes out on Friday. Should we make the seudah with music and dancing on Thursday night?
It is not forbidden for one to have a seuda, dancing and music the night of asoroh beteves (Thursday night). However, you carefully worded your question as whether or not it is proper. This indicates to me an interest to arrange your party only in a time when it is lechatchiloh permissible and there is no basis to be stringent. There is reason to avoid having the seuda, dancing and music the night of asoroh beteves.
Although the Shulchan Aruch (550:2) permits one to wash oneself and the other inuyim on the three fast days (other than Tisha Beav and Yom Kippur), still, one who is particularly scrupulous regarding mitzvos should consider the minority opinions and refrain from all the inuyim just as if it were Tisha Be’av (m.b. 550:6).
The Eliyahu Rabbah (550:2) quotes the Sheloh as suggesting to all those capable to begin the fast of Asaroh beteves and all other inuyim from Sundown instead of alos hashachar just as on Tisha Be’av. So, although such behavior is not normally noticeably practiced, there is basis to avoid making the party on Thursday night.
Unless there is some pressing concern, I recommend that you arrange the birthday party as a melave malka on the 12th of Teves. It is advised to offer some words of appreciation to Hashem for all He has bestowed upon you and some words of Torah. A proper seuda birthday/melave malka with song, music, praise to Hashem and divrei Torah is a much greater seudas mitzvoh than a birthday party the night before the fast of Asoroh Beteves.