I'm not sure which airline you can use as the agreement varies. Note that some are accepting only E ticket and some are paper ticket. Its a big confusion right now.
There is no particularly big festive season around that time. There maybe a few long weekends, i.e. Fri/Mon holiday but since you are avoiding the weekends you should be fine.
School holidays may start as early as mid Nov, so my recommended timing would be mid Oct to mid nov. Early Oct is bad as its a big Chinese holiday and 1 billion people will be trying to get out.
Here's a break down of your routes
BKK-HKT-BKK
I've not flown this but I think your only option is TG only and they have 4 flights per day so it should be fine if you start from the first flight. Most passenger on this will be tourist and since its off peak the load shouldn't be that bad
BKK-SIN-BKK
I'm not sure if your airline has agreement with SQ (you'll need to get ID and not ZED tickets for them) , because 1/2 the flights available will be on them. BKK-SIN runs will be tight if you can't take SQ. The route has heavy load of business, tourist, etc ,etc
If you can do both TG&SQ I think you'll be fine
SIN-KUL
These used to be "Shuttle" flight or thats what they call it. You can buy a firm ticket or standby ticket, you take a number and queue for the next flight which could be MH or SQ, this system allows for seats at most times. I'm not sure they still have that because budget airlines have taken a big chunk out of their business of this sector. While loads are heavy (we use to fly 1 way in the morning for meetings and back the other way in the evening/afternoon), but you should be able to squeeze 1 seat in or 2. Early morning flight (before 8am) will be mainly business/lawyers. You have a chance on that because its too early for some corporate people. 8 to 10 will be packed, after that its lighter again. and the same for 4pm-8pm stretch when people are done for the day and on their returning flight.
HKG-TPE-HKG
This is my favourite route that I call it the bus route. I can take CX/KA/CI/BR. Frequency, is almost 20-30 minutes. The only problem is that the flights depart from 2 different terminal and you have to run from 1 terminal to the other to checkin for the next available flight. Unless you don't mind taking one that is 1 hour or 2 later, you can just sit there and wait. There is limited flight that flies from Taiwan to China (political stuff). So every taiwanese needs to go to China via HKG, and since almost all taiwan factories have shifted to China, there is an abundance of flights doing this route.
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