Ah.
"There was basically no difference in the speed gap between ISP Gateway modem and DAP-1360. I have tried various Channel numbers manually, but found the signal to be weak and have disconnections. That is why Auto Channel Scan is checked."
That could be your problem. You have tried lots of things. The only thing left that I can think of (other than to disable QOS (WMM). Try and reset the modem with the button on it, to see if you can start over from scratch, and disable WMM. Then setup the rest of the unit.
Why not try and go for standard 802.11g to 802.11n only, select only 20Mhz width, select a channel that is free from others by using inSSIDer, disable auto channel scan, and select a channel that should work. Select WPA2 with AES. Thats it. And then do speed tests with the laptop of whatever you are using close to the DAP-1360 for a strong signal. This should get you to 40-60Mbit/sec and that is an acceptable speed.
I do not understand why you can not disable QOS, WMM, I could do this on my unit with both v3.02 and 3.04 firmware. Must be something blocking you from preforming that task, as I know in some configurations some options are greyed out. So you have to do some tasks in the right order to be able to select the correct settings, a bit tricky but should work.
If none of the above works, buy a TP-Link 5dbi antenna or 8dbi antenna for stronger signal. There is in general a problem with some phones, laptops etc, that have bad antenna placement, this prevents high signal quality, even though you have good signal strength. Also if you have a bad antenna in the laptop or signal is not that high, a larger antenna (and stronger) on the DAP-1360 will make the unit work better. Plus the antenna I am referring to is not expensive and works well. If this fails to, I can not help you out more.
The only thing left is to see if the "Gateway Modem" is the problem, how to test this ? well you could try and connect another laptop of unit, and transfer a file over the LAN to check what speed you get, at least then you will know if the LAN part of the network is working full speed. As far as I can tell what you have done should work, only thing I do not like is that WMM is enabled.
Hope you get things sorted.