It's weird to doubt Phuket immigration. It's their job to tell you how and went to get your visa. Nobody here can give you better info then the immigration officer. Brandon is probably right, likely you got the under consideration stamp.
Did you choose Taipei because you want to stay there a week and visit? If not, just take the bus to a neighbor country like Laos or Cambodia, it's much easier to get tourist visa there. Taipei is not a common place for visa runners, but I'm sure some did it this year. Good luck!
You can get it as many times as the embassy in Singapore will allow you. And no problem getting back in. Worse case you call an agent while in airport who will bribe your way in for around 8k baht. This is what we call the VIP service with agent which guaranteed you get in.
OR... after getting the 90 days, you can extend for 1 year with an agent for around 25k baht (without showing 400k). That's the cost for using an agent. You can by-pass the financial requirement with agent. It's up to you.
What visa? A tourist visa? You can extend tourist visa and visa exempt by 30 days at nearest immigration office. If you already did it, I suggest you wait closer to November 17 to leave. Are you married? If so, yeah, go to Savannakhet around November 15 and apply for 90 days Non-O (they are not asking for 400k baht). And then you can extend 60 days at immigration office near when the 90 days expire.
For now, thankfully, single entry Non-O based on marriage at Savannakhet doesn't require 400k. It give you 5 months (60 days family extension), so not too bad. My friend will be in Savannakhet this Friday or Monday and he will try to pay the 13k baht bribe to get the Multi Non-O. Rumor has it that it's the current price with a "fixer/bribe".
You can't do the 90 days report online if it's your first 90 days report. It must be the case because 1 days late will not be denied if you try to do it online. You can do it online up to 7-10 days late.