Yeah we do A/T, manually increase throttles to N1 at 40 then press the TOGA buttons. Forgot what VNAV does, helps when you turn on the A/P or? I think we call out VNAV either at 400’ or 1000’, I forgot.
On the line, we ARM the A/T, VNAV, and LNAV, if applicable to the clearance, when the takeoff data is loaded (uplink). That can happen as early as at the gate, if the loading has been closed out by ramp and customer service, and must happen before we do the Before Takeoff Checklist. Your picture showed you on the runway but the FMA was still blank. Obviously, reposition to that point is common in the sim.
TOGA does as you say, if the A/T is armed, and puts both pitch and roll into TOGA which will command the pitch for the initial climb on the F/D and command liftoff track on roll.
Roll mode will switch to HDG or LNAV, depending on the mode that is armed, no later than 400'. Some RNAV departures engage LNAV much earlier. See the Rwy 1 departure from DCA. LNAV engages immediately on liftoff to turn away from the prohibited area.
400' is the minimum autopilot engagement altitude per the Limitations section of the FM but it's really not a lot of help at that point. Easier to hand-fly the acceleration segment.
The acceleration schedule is programmed in the Takeoff 2/2 page and CLB page. That has your thrust reduction altitude, acceleration altitude, and engine-out acceleration altitude. VNAV will command a pitch attitude to accelerate to the the CLB page speed restriction while protecting the max flap speed for the current flap setting during clean up.
We set either an L1 or L2 climb, usually and L2 climb. That puts a 250kt limit on the CLIMB page L1 and clean maneuvering speed to 3,000' above the runway. That's an NADP2 departure profile. As flaps are retracted, VNAV will command a pitch for clean maneuvering until 3,000' above the runway then will command 250kts until we select, and execute, ECON CLIMB. In ECON CLB, VNAV will command a climb speed based on the CI (Cost Index) entered in the PERF/INIT page during setup. We use CI 10 as the default then increase from there if CI 10 puts us behind schedule.
You might find it interesting that we had a failed TOGA in October. I hit TOGA, several times, but it didn't take. It was my takeoff so I alerted the PM and emphasized for him to SET thrust. (Just like my 15 years flying DC8s and DC9s) After liftoff, the A/T modes operated normally. On the next takeoff, TOGA operated normally. Never did figure out why it didn't take.
My normal procedure to set takeoff thrust is to push up to 40%, N1 until the engines spool, push to 70% N1 to monitor normal acceleration, then push TOGA and follow the thrust levers as the A/T sets the scheduled T/O thrust. In this case, I pushed up toward takeoff thrust, when TOGA didn't engage, so that there wouldn't be a delay in achieving power and the PM would only have to fine tune it.