I have added some details to @Nevin Williams' answer. In fact his answer led to the solution. But, what were the outcomes, I also wanted to share those.
Here the building wiring was creating the problem and it was solved by using external CAT-6 FTP wiring.
The exact answers to the questions asked go as following:
Should I really spend money purchasing Cat 6 instead of using the in-building pre-installed normal telephone line wire?
Ans: Yes if the test shows a difference, as said above.
Will it really improve things?
Ans: Yes it will surely improve.
I am on the 3rd floor, so will it work at a length longer than 50 feet?
Ans: It (CAT 6 FTP) has already worked perfectly for a length of 51 meters (~167 feet) to carry a DSL line.
If yes will I be using only two lines—one pair out of four—from that Cat 6 cable?
Ans: Yes, only one pair.
Also can someone suggest what’s going wrong here—if anything—and what would be the fix?
Ans: well, After the change of this cable the stats in question have been changed to the following(which show an improvement and a stable connection, as we can see here about SNR and attenuation):
Line standard: ADSL2+ Channel type: Interleaved Downstream line rate (kbit/s): 10239 Upstream line rate (kbit/s): 510 Downstream SNR (dB): 19.8 Upstream SNR (dB): 29.8 Downstream line attenuation (dB): 12.5 Upstream line attenuation (dB): 5.5 Downstream output power (dBmV): 0 Upstream output power (dBmV): 12.6 Downstream CRC: 0 Upstream CRC: 0 Downstream FEC: 11 Upstream FEC: 0
Also as an important note I would like to mention that, I did get 10 Mbps sometimes with internal wiring, but it used to decrease Downstream/Upstream SNR to ~6dB, while its now ~20dB and ~30dB, respectively. That's the big gain, that really solved the problem of disconnection.