White offers a draw. Does he then punch his clock and Black loses time while he considers it? Or does an arbiter stop both clocks?
In High School matches no arbiter is involved.
Typically the person offering the draw wouldn’t hit the clock. If they did the other player can consider both their move and the offer at the same time and if indecisive make their move and hit the clock and just say they’re considering it.
I seem to recall move, offer draw, hit clock, and reading up, that does appear to be the usual sequence.
Yes, the process is to make your move, then offer the draw, then hit the clock. The other player can consider the offer on their own time.
@glee recently told a story of offering a draw on the first move of a speed chess game, and his opponent spending 20 of his total 30 minutes considering it before accepting.
And in the first Kasparov-Deep Blue matchup, at one point Kasparov took some time pressure off of himself by offering a draw, because he knew the IBM team would eventually decline, but that it would take them a while to make that decision. With the time pressure eased on him, he eventually won.
I’ve read your post a few times and I don’t fully understand it - if you offer a draw without hitting your clock, then your opponent can think about it on your time, which is worse. Hence the usual procedure, already described.
I believe if the player who was offered the draw makes their move and hits their clock, that is taken as declining the draw offer, and if they later decided they did actually want to agree a draw, they would have to make a fresh offer themselves. Of course, the sooner they did so, the more likely the original offeror would agree. But they would be under no obligation to do so if a move had been made since their offer. On the other hand, if a draw is offered and accepted, obviously the offer cannot be withdrawn at that point.
I agree with pulykamell and borschevsky - the sequence is move, offer draw + then press clock.
Some anecdotes from tournament play…
In a complicated position, one player muttered “are you interested in a draw?”. If I said yes, presumably they were going to reply “Well I’m not!” and play on. 
So I muttered back “Are you offering a draw?”
No reply - so we played on … and I eventually won. 
Another player offered a draw without first moving (this has happened several times.) Again, presumably if I turn down the draw, they would choose a particular move.
So I reply “I’ll think about it.” This establishes that I have been offered a draw and leaves their clock ticking, so they will in due course have to choose a move without knowing what I think of the game position.
It’s a bit embarrassing when an opponent makes a move, offers a draw and then sits back without pressing their clock. 
Obviously they are waiting for your decision and have simply forgotten the clock press - so they risk losing on time…
My policy is to discreetly gesture at the clock, so they can press it.
A team-mate of mine decided to pretend to be lost in thought (hands on head, hunched over the board etc.) and ultimately won the game on time. This led to an argument … but the result stood. (It’s your responsibility to press your clock.)
When playing in international tournaments, there is a chance that your opponent will not speak English. (Mind you, I think it’s unlikely - I remember watching a post-mortem analysis between two Grandmasters. One was Icelandic, the other Hungarian … and the both spoke perfect English!)
So I use the word ‘remi’ if I don’t think ‘draw’ will work. This is widely accepted.
I was playing a venerable International Master in the Netherlands once and reached what I thought was a level position. So I moved, offered a draw and pressed my clock.
No reply - not even an acknowledgement.
After a bit my opponent moved - and it was a move without ambition. He then went for a walk.
I wondered if he had heard my offer, so asked the arbiter if he would ask my opponent if he had heard my offer.
The arbiter came back and said “He heard you!”
The game meandered on until my opponent proposed the draw. (Apparently he thought only the higher-rated player should make the offer!)
In the last round of another event, I only needed a draw to qualify for the FIDE Master title.
I achieved a large advantage, so offered a draw. (The title meant a lot to me, whereas I wouldn’t win any more money by winning.)
My opponent declined … and lost.
Afterwards he explained he needed a win for the FIDE Master title.
I apologised as I would not have suggested a draw if I’d known.
This seemed to happen to me quite a bit back in high school tournament play (with a draw offer, or during the normal run of play after a move). I think the first time I tried to let it run, but discovered I felt so distracted by/guilty about his clock running that I finally said something. After that I just told them.
The FIDE rules (pdf) say:
A player wishing to offer a draw shall do so after having made a move on the chessboard and before stopping his clock and starting the opponent’s clock. An offer at any other time during play is still valid but Article 12.6 [nb: That rule prohibits you from distracting or annoying an opponent, and the unreasonable offer of a draw is explicitly mentioned as an example] must be considered. No conditions can be attached to the offer. In both cases the offer cannot be withdrawn and remains valid until the opponent accepts it, rejects it orally, rejects it by touching a piece with the intention of moving or capturing it, or the game is concluded in some other way.
So my interpretation would be that the draw must be offered before hitting the clock, i.e., on the time of the player making the offer. This does not act to the offeree’s disadvantage, also because the touching of a piece with the intention of making a move counts as an implied rejection of the offer. So you don’t even need to say anything if you want to reject the offer.
The “An offer at any other time during play is still valid” phrase is a bit cryptic, as it would imply that you can also offer a draw on the opponent’s time, contrary to what the rules said before. But since the opponent can simply proceed to make a move, and that counts as a rejection of the offer, there’s no disadvantage caused to the offeree. Potential for abuse is reined in by the rule that prohibits causing distraction or annoyance.
When I was still playing over the board, this annoyed me a lot when it happened (infrequently fortunately). The moment my opponent made their move, I focused completely on my direct response and consequences for the remainder of the game. Not the best moment to have my train of thoughts disturbed by hearing a draw offer.
I think the rule could be worded more clearly, but the intention of it is for the sequence of events to be: move, offer draw, press clock. That way, the offeree is aware that there is a draw offer to consider as well as their next move. As opposed to, as mentioned by Art_Rock, pressing the clock and then interrupting your opponent’s train of thought with a draw offer. But I’m sure there is no expectation that you should refrain from pressing your clock having made your offer - there is no reason to let the opponent use your time to think about it, that wouldn’t work at all.
Just once in my chess career, I offered a draw without first making a move.
This was because my opponent was horribly short of time (they had 8 moves to make in a matter of seconds.)
However I had just blundered a pawn - and I knew my opponent was a pretty quick thinker.
Anyway they took the draw. Physically making 8 moves and pressing the clock is tricky!