Hi - I have done the WCT 4 times, the last this past May - first on, first off.
1. How cold does it get at nights? I'm worried about bringing warm enough camp clothes. My sleeping bag is rated to 35 degrees.
You will be fine with that sleeping bag temperature range, especially in the summer and early fall. The only consideration is moisture control. Everything seems to get damp even if no rain - the humidity is considerable.
2. Is the trail/beach buggy? Do I need to worry about bug spray?
The trail can be a bit buggy, but you will be moving enough that it is rarely an issue. The beach is fine except at the odd campsite when sand flies come out. They bite. I would take a little deet or comparative just in case.
3. Trekking poles or not?
Absolutely. Not only do they help on the trail in the muddy areas (you will likely be doing about a KM an hour in these areas (about 22KM of it) but are valued on the sand and tidal shelf. You will not be disappointed with the additional support.
4. I plan on one pair of hiking clothes and one pair of camp clothes (long johns) with one extra pair of socks and briefs. Then I will have my DriDucks rain gear. Does this seem sufficient?
It seems sufficient to me. You will not be doing any bushwacking (although I did and that is a loooong story) so the DriDucks should be fine. Extra pair of briefs is a good idea but I might bring 2-3 pairs of socks because of the potential for wet feet.