I just actually realized, when I was checking your traceroutes and comparing it to mine, your connections are routed through different line in NA than what my are.

Here's a traceroute from connection/computer that CAN log in to game:

Tracing route to traceip.ffxiv.com [202.67.52.81]
over a maximum of 30 hops:

1 17 ms 3 ms 3 ms 78-27-64-2.bb.dnainternet.fi [78.27.64.2]
2 3 ms 23 ms 3 ms lah1-tr1.dnaip.fi [62.78.105.96]
3 12 ms 12 ms 12 ms hel5-tr3.dnaip.fi [62.78.107.126]
4 12 ms 12 ms 12 ms hel1-tr2.dnaip.fi [62.78.107.124]
5 13 ms 15 ms 14 ms hel2-tr1.dnaip.fi [62.78.107.9]
6 12 ms 13 ms 13 ms tuk2-tr1.dnaip.fi [62.78.107.57]
7 17 ms 22 ms 22 ms 10gigabitethernet1-3.core1.sto1.he.net [194.68.1
28.187]
8 37 ms 38 ms 48 ms 10gigabitethernet3-2.core1.ams1.he.net [72.52.92
.45]
9 45 ms 49 ms 47 ms 10gigabitethernet6-1.core1.lon1.he.net [184.105.
213.245]
10 115 ms 114 ms 111 ms 10gigabitethernet7-4.core1.nyc4.he.net [72.52.92
.241]
11 180 ms 177 ms 172 ms 10gigabitethernet5-3.core1.lax1.he.net [72.52.92
.226]
12 197 ms 203 ms 201 ms pacnet.10gigabitethernet2-3.core1.lax1.he.net [2
16.218.223.206]
13 299 ms 302 ms 304 ms gi6-0-0.cr1.nrt1.asianetcom.net [202.147.61.166]

14 308 ms 317 ms 301 ms gi1-0-0.gw1.nrt5.asianetcom.net [202.147.0.178]

15 301 ms 300 ms 304 ms squareco.asianetcom.net [203.192.149.210]
16 312 ms 304 ms 305 ms 61.195.56.129
17 291 ms 295 ms 298 ms 219.117.144.2
18 295 ms 297 ms 298 ms 219.117.146.149
19 299 ms 300 ms 297 ms 219.117.146.129
20 300 ms 301 ms 302 ms 219.117.146.38
21 293 ms 308 ms 293 ms 202.67.52.81

Trace complete.
As you can see, your connections are using seabone to go through NA, I'm using core1. Could maybe be a problem somewhere in NA too? Seems like your timeouts start after entering NA networks.