Yes, at least one audio stream must be present regardless if your content has audio. There doesn't need to be anything actually in the audio stream, it just needs to be there.
FCP has a video filter which can READ and display timecode either from the source clip or the sequence timeline, but can not generate the audio pulses. You'll need a real timecode generator to create the correct SMPTE LTC pulses, which can be output as an audio signal. A reliable, moderately priced brand is Horita, but there are others. Many devices also can insert a burned in timecode window.
I would again consult with the event people to see what kind of timecode reader they are using, and if they have any suggestions about generators which have given them the best results in the past. Maybe they have a generator you can borrow.
Depending on the sophistication of the TC generator, you can select a specific start time, jamsync to a specific video reference point, drop/non-drop, frame rates, gen-lock and EBU TC for PAL. You can capture the output, both audio and video (for your reference), into FCP.
Use different TC start times (e.g., 01:00:00:00, 02:00:00:00, 03:00:00:00...)to distingush the DVDs for the event folks.
Your DVD layout is probably fine. If the Menu is indeed in Menu-Space, and the button click to play is what's necessary for the event, then OK. Double-check with them on whether that 's what they want/need, or if First Play should be to the Title. No need to set an end to a Chapter. Chapters end automatically at the start of another Chapter, or at the end of the Title.
Let us know how things turn out.
- TV