Explanatory 'showcase' answer provided by OP: All three of the existing answers here made great contributions to solving the puzzle (@Graylocke, @samm82, and @berkeleybross - to whom I have awarded the checkmark for being first to find the final answer), but none of them quite described the intended solution path in full (without making any assumptions), which for interest and completeness I present here...
Clearly the first thing to do is solve the 16 clues - everybody worked that out :) But what are the answers? Some of them look like they could be ambiguous...
For instance, 'French, soft, creamy cheese' could be Brie or Camembert (or other less well known cheeses), and there are several 'Major German automobile manufacturers'...
But there's something noteworthy about all of the clues:
Every clue is exactly four words long (taking 4'33" as being 'a word', here...).
This uniformity suggests that perhaps there is something uniform about the answers too. Specifically:
All of the unambiguous clues seem to lead to answers of precisely four letters. So perhaps all of the clues should do so.
This yields answers that look like this:
American composer of 4’33”. CAGE
Capital city of Ukraine. KIEV
Monty Python member, Eric. IDLE
French, soft, creamy cheese. BRIE
Bible’s first murder victim. ABEL
French fashion designer, Christian. DIOR
Miami professional basketball team. HEAT
Ctrl Z in Windows. UNDO
Norse husband of Frigg. ODIN
Lenin’s federal socialist state. USSR
Tony Shalhoub detective series. MONK
First South Korean President. RHEE
'Hold the Line' band. TOTO
Part of the brainstem. PONS
Major German automobile manufacturer. AUDI/OPEL
2003 Colin Farrell movie. SWAT
So it looks like we're onto something!
But at this stage we're still not sure which of the major German automobile manufacturers is the intended one - both AUDI and OPEL can fit. We clearly need to find a way to disambiguate this. So what else do we have in the puzzle?
Well, very prominently, we have the chessboard. Moreover, in the text there is mention of one of the attendees writing on the chessboard. And 16 answers of 4 letters makes 64 letters in total - is it a leap of logic too far to suggest that we should assign one letter to each square of the chessboard?
How should we do this? The only perceivable further hint here is that:
The chessboard has a specific arrangement of pieces upon it. Furthermore, in ordinary chess algebraic notation each piece is represented by an initial: K for King, Q for Queen, R for Rook, B for Bishop, and N for Knight (as K is already assigned to the King). Traditionally in this system, pawns are indicated by the absence of a letter. However, in the old descriptive notation system which it superseded altogether at the start of the 1980s, Pawns are indicated by P.
So what does the chessboard look like if we make use of these? This:
Counting up the letters here and in the answer shows an exact one-to-one correspondence of K's, R's, B's, N's and P's (there are no Q's) - if we use OPEL instead of AUDI.
This now suggests a strategy:
We should fill the board as far as possible as we can using words that can only fit in one position. As per a regular crossword it makes sense to attempt this using a straight-line arrangement in the first place and see if this gets us anywhere (rather than, e.g., the adjacent-letter approach of a Boggle grid) - especially as thus far there has been a sense of uniformity to the puzzle, with its four-word clues and four-letter answers.
KIEV, MONK and PONS (only one possible place):
BRIE and ODIN (now only one possible place):
ABEL, UNDO, RHEE and OPEL (knowing that for an exact fit, each letter is used only once):
This is as far as we can go without ambiguity. And it's at this point we can spot a solid pattern:
So far, all of our clues have been written vertically downwards onto the chessboard, either fully in the top half or fully in the bottom half:
The only legal way to arrange the remaining words in the board is to fill them out likewise. What's more, we can see that if we consider first the top tier of words, then the second, left to right, the answers entered so far appear in the same order as their questions appeared in the list - and DIOR and USSR are in appropriate positions to satisfy the two remaining R's on the board...
So let's enter the remaining answers according to the same system to produce the final grid:
What do we now? Well, likely we're looking for a final message concealed within the completed grid. And a little trial and error leads us to a solution:
If we only read the white squares on the grid, the message is revealed! CIA HIRING. LEAVE ROOM TO SHOW INTEREST.
(NB This works only if KIEV is used instead of KYIV, another possible spelling of the capital city of Ukraine - if we had worked with KYIV from the start instead, we would be able to disambiguate that at this point, as 'HIRING' is a real word while 'HYRING' isn't...)
So it appears that:
This whole scenario is actually a CIA hiring event! And anybody who managed to find the message should have left the room early to express interest in taking up the offer, having demonstrated their ability to interpret the chessboard cipher. No doubt there would have been somebody stationed outside the room to intercept anyone leaving and clarify their intentions (the CIA don't just want to go recruiting folks who are nipping out for a toilet break after all!).
Of course, within the puzzle we never worked out any of this, so we go about our daily business none the wiser, a little puzzled but oblivious to the opportunity we just missed out on. Meanwhile, the chap who did leave early - likely the same person who was writing directly on the chessboard - is on his way to becoming a CIA operative!