I am looking for advice regarding a situation of my friend:

He found a car he was interested in purchasing, but at the dealership
it turned out the "new" car had 100 miles on it. It was shoddy, dirty,
and missing the back logo. Nevertheless, he wanted to purchase. He was
financing through the dealer. The dealer was not able to process his
finance application, and forced him to sign a document saying he would
buy if they got financing, if not, he would return the car. He also
signed a promissory note, a very simple 5 sentence once stating he is
finanically responsible for the vehicle, but this note did not
explicitly seem to be a sales agreement in any sense.  And the
dealership took a deposit on a credit card.

He drove the car out that night, but immediately discovered a slew of
other problems no doubt resulting from the high mileage on the car and
fact that it had an inspection stick from October, indicating it was
sitting for a while.

My friend expressed his concern to the dealer, and asked for them to
accept the car back and cancel the deal, especially as there was no
financing in place arranged by the dealer. My friend drove the car
back, but the dealer refused to formally cancel the deal. He left the
car there.

Here is the current situation:

The car resides at the dealer, although they said as my friend left
that "the car is still yours";

My friend has a document only stating he will buy when the dealer
arranges financing, however the dealer is silent and there is no
financing 4 days later;

There was a temporary tag issued and temporary registration card - I
do not know the implications of these docs.

My question is: What steps could the dealer realistically take to
"force" the car on my friend? My friend intends to leave the car
there, and if the dealer does not contact him, he will go about his
business. It seems to me (I am in law school now) that the dealer will
have an issue trying to collect on the promissory note as the car was
voluntarily returned by my friend. Additionally, there is no
financing, and I can't see how the dealer would ram this down my
friend's throat and force him to sign the loan.  My friend never paid
for the car.

However, I was concerned that the dealer can't just absorb the vehicle
back into inventory unless my friend possibly signs something - does
anybody know who owns the car now, what the DMV has in the way of
information, etc.?

What do you think about the issue of financing? When I read the doc my
friend signed in that regard, it was pretty thin - basically 4
paragraphs stating that the dealer will arrange financing, if not my
friend has to return the car. The dealer has not been in touch re: any
financing, and my friend only applied for Honda's financing, so they'd
have to force his hand to fill in other financing applications if they
wanted him to get financed elsewhere. But the whole deal was
contingent on the dealer coming up with financing which didn't happen.

You are in law school?  First, how did the dealer "force" your friend
to sign anything?  How do you know exactly what your friend signed? 
Does he have copies of everything he signed?  The dealer could go to
court and sue for specific performance.  It is not a valid defense
that "I left the car on the dealer's lot."
That becomes abandonded property.  I have heard of this before; why do
people think they can cancel a sale by leaving the car on the dealer's
lot?  Bad move.
Your friend (couldn't be you could it?) should have test driven the
car - that's the time to find things wrong - not after signing for the
car. You even say the carlooked in poor condition.

This needs to be handled in a business like manner.  If you are in law
school there must be someone doing pro bono work that will review the
documents that were signed and determine what is going on.