About the first question, if a Yisroel can perform shira on his own korban:
According to the Meishiv Davar chelek 4 siman 11 Leviim are simply hired by Yisroelim to perform shira (therefore he says that if a Levi says that he doesn’t want to do shira there is a hava amina that he wouldn’t have to, if not for a pasuk that says otherwise). If so if the Yisroel himself wants to not hire all the Leviim, and wants to do his own shira for his own korban, maybe he can. He only can’t do shira for another Yisroel (or Levi or Kohein) since the Leviim are the ones who get the monopoly on being hired for shira, and the pasuk is קובע that they and they alone are the ones who are hired to do shira for everyone else. But here he isn’t hiring anyone so maybe his shira is acceptable.
About the second question, if the shira of a Yisroel is kasher bedieved (when not for his own korban):
Relevant is the Minchas Chinuch 389:3 who says that there is no issur for a Yisroel to do shira in the Beis Hamikdash. My question is if there is no issur and it's kasher, or if there is no issur but it's still pasul. And even if you say not like the Minchas Chinuch, it's still possible that it's kasher bedieved, even though there's an issur.
[edit - I have since seen the Rambam who says that there is no shira for a korban yachid. If so it would seem to be a davar pashut that the 'baal hakorban', which means any yisroel since we are dealing with a korban tzibur, can't do shira at all, since it's always for a korban tzibur. If any yisroel can do shira then where is it chal the din that leviim are the ones who do shira? So the first question would seem to be answered. The second question however does remain.]