PJ - Pira has answers for you. The (serviceable) 737 system will not allow the Autothrottle to increase either TL position after 2 seconds from touchdown. I am not sure how long the 'Retard' function stays active (I assume 2 seconds?), but even if TL was 'automatically' advanced within those 2 seconds for some reason,
a) There is a direct 'Visual' clue of lever movement, and further
b) Reverser cannot be selected
Regarding A/T function on start-up, yes, if A/T is selected the engine will accelerate to match T/L position. It was, some time ago, 'SOP' in one airline to select A/T before start and I was with a Captain who had inadvertently selected the g/a thrust button during start-up which widened the ground-crew's (and my eyes somewhat). It is, I believe, now back in place in the 'new' Boeing 737 checklists (A/T select, not TOGA
).
As far whether both T/Ls need to be at idle to select ANY reverse, I know not, it is not an area I have explored but perhaps we can have someone try (before start up, I suggest, with notice to groundcrew!!). Since a jammed throttle lever would give this problem I am guessing that only the required lever has to be at idle. I have not yet seen an answer to whether the 737 can have one throttle advanced manually with one in reverse or whether they are locked - again not something I have experienced. Perhaps again some kind soul could check before start-up?