All the hedge knights that I can think of introduced in the series so far - Ser Illifer the Penniless, Ser Duncan the Tall, Ser Pate of the Blue Fork, Ser Shadrich of the Shady Glen, etc. - don't have last names, at least that we know of. The one exception I recall is Ser Creighton Longbough, but there is no mention of a House Longbough on AWOIAF, so this may be a name Ser Creighton has invented, not a House name.
I can't find any information that specifically precludes hedge knights from having House names. But (minor spoilers AFFC)
"Knights they are," said Petyr. "Their gallantry has yet to be demonstrated, but we may hope. Allow me to present Ser Byron, Ser Morgarth, and Ser Shadrich. Sers, the Lady Alayne, my natural and very clever daughter ... with whom I must needs confer, if you will be so good as to excuse us." The three knights bowed and withdrew, though the tall one with the blond hair kissed her hand before taking his leave. "Hedge knights?" said Alayne, when the door had closed.
Which seems to suggest that hedge knights as a rule don't have House names.
So my question is twofold:
Is there a reason that none of the hedge knights we've met appear to have House names (Ser Creighton possibly notwithstanding)?
Could someone from a noble House become a hedge knight and retain their House name?