Yes, it is possible to stop an Ephemeral being with regular (non-Bane) attacks, but they're not as efficient as they are against mortals.
Do they have to make a reflexive roll to stay conscious?
No, but the rules are not written as clearly as they could be on this topic. The rule you are referring to can be found in the WoD core book Bashing Damage section on p. 174. "Once all your character’s Health boxes are full and the rightmost is occupied with a slash, begin making Stamina rolls to see if he remains conscious.” However, Health (with a capital H) is not a trait in play here and the characters described in this book are PC mortals. In the God Machine Chronicle core book where Ephemeral Beings are defined, on p. 220 we see that “Ephemeral beings don’t have Health, but measure how intact their Twilight form is using Corpus.” (Not the question, but related: note here that “Corpus boxes don’t have wound penalties associated with them.”)
We get some more clarification on the rules in Blood and Smoke which, like Demon, was one of the first books published after the Rules Update. On p. 181 we see, “If a mortal’s health track is filled with bashing damage, his player must make a reflexive Stamina roll each turn for him to remain conscious.” Other rulebooks for supernatural beings explicitly specify for the character type if you need to make rolls to prevent unconsciousness with a full bar of Bashing Damage. Vampires: No (Blood and Smoke p. 181). Mummies: No (Mummy: The Curse p. 144). Promethian: No (Promethian: The Created p. 162). Changeling: Yes (Changeling: The Lost p. 173).
Do the subsequent Lethal attacks then act as Lethal?
Subsequent Lethal attacks are downgraded to Bashing as usual first. Then, as the Corpus boxes are all full with Lethal Damage, they upgrade Bashing Damage to Lethal. So, yes, after a fashion.
If that track gets filled again, do they take Aggravated damage?
They would except that “Ephemeral entities that lose all Corpus from lethal or aggravated wounds explode into a burst of ephemera” (GMC p222). At this point you have temporarily stopped the Ephemeral being.