one simple, one possibly more complicated.
13a surpassed unfashionable 4, we hear(8) OUT*O*E .The E may be questionable.
6d To put a girdle round the earth in quatrain or ballad(5) I*O*B
This yields me nothing. 4a He'll produce villanelles initially whenever that's right,without signs of hesitation. I think this is VERSIFIER but it gives the first I for 6d.
Help needed to sort out this last bit.