Stubs are relatively short(varies around 1-2 meter),
In a complex, long distance 485 setup 1-2 meters is NOT "short". I run 2 cables into my units just to have the stub only a few inches. I only run about 50 ft at 460kb

the cable I have used is Cat5 and unshielded.
This may be your problem. While the differential signal does not require shielding, the common mode does. While your transciever may be able to withstand a lot of common mode, it can only process a few volts.

Terminating resistors are 120R and resistive(I also tried 100R) and located at the extreme ends. RS485 chips used at nodes are of 1/4 UL.
Use the 3 resistor setup in either or both ends. I do not know this, but if you have a "dip switch activated" terminating resistor, just one dip switch set wrong can cause pure havoc. for that reason, I always use automatic termination.

- If I cancel the terminating resistors or if I use a larger resistors, distance drops dramatically to 100-150 meter.
Read the app note, it states exactly that.

- If I use my converter with its external power, result is even worser than that of powerless(port powered) mode. This implies that bus needs biassing, but every overture I made resulted with no any good..
Get rid of the converter. Make a simple '51 board with a 2 UART derivative that receives the data, verifies it and then transmit on the other channel. While plug in converters usually work well for point to point, they are lousy for multipoint. Re biasing, see above

- Even though I use a single node on the bus, I still couldn't provide a distance farther that 400 meter. This stops me thinking that I am suffering from overloading.
If you leave the stubs in, just disconnect them it malkes no big difference. If only 2 terminations (make sure, do not go by what you "know") that is not the case any transmitter can handle 128 1/4 load receivers and 2 terminations.

- I had omitted the unused cables in cat5 that should have been ended with 100R resistors to GND. I will do it soon.
Do you think it is that why such a poor result I got?
I doubt that very much.

And I planning to descrease the data rate to 9.6Kbps from 19.2K and see if it will do any use.
any incorrect 485 setup will perform "better" at lower rates. If you went to 300bps, you might even get data through without termination.

Erik

PS you DO wait 0.6 bit time or more before turning the bus around after last byte I hope.