Hi -

1) Absolutely take the tunnels. Once past security, it is about a 15 minute walk from T4 to T6; maybe 20 minutes from TBIT. No moving sidewalk, but there are some interesting historical murals on the walls.

2) With a 2.5 hour wait, you should be able to enter the AA Flagship Business Lounge in T4. There you can take a shower, grab something decent to eat and/or drink, good wifi, etc. If you are jet lagged, the unlimited Red Bull might be the thing. Note that they neither show AS flights on their departure screen screen nor announce them, so you have to find out online or on your mobile app.

Again, about 15 minutes from there to T6 via the tunnels make sure you are at the gate at least 30 minutes before departure, which means leaving the Lounge by T-45.