the weapon affects the respective troop type
the +500 is for your hero strength the +3 is added to the attack value for each swordsman attacking with the hero being mounted or unmounted makes no difference (even if your hero was unmounted it would at most only add +3 to your attack value, since he is only 1 troop)
well i tried to cover all posts, as you mentioned Horses and Swordsmen are foot soldiers, so wouldn't really apply
in the example you gave the "short sword of the swordsman"
it gives you 500 fighting strength for your hero & it also gives
+3 attack value to every swordsman accompanying your hero (your basic swordsman would have an attack strength of 68 instead of his usual 65) so 100 of them would give you 6800 instead of 6500 etc
as mentioned if your hero were unmounted (i.e. also then a swordsman) he might also get +3 added to his attack value + the 500 from him holding the weapon
the hero always gets the 500 base strength of the weapon, but only the respective troops would get the +3 extra, so wouldn't give +3 attack to horses or phalanx etc
as mentioned if your hero were unmounted (i.e. also then a swordsman) he might also get +3 added to his attack
No. why would a hero got it? Why is hero a swordsman and not a phalanx?
Everything else is correct.
he carries a sword it could be argued he's a swordsman 
if he carried a spike he may become a phalanx
The attack bonus is added to the hero power, thus, if the is mounted, you get +3*swordsmen mounted attack
For a simple experiment run the ingame calculator with hero power 0 and whatever set of attack/defense (but with at least some troops of the weapon type), you will notice the defense value changes if you put your hero on horseback or not (because of the now different proportion of mounted/unmounted attacking value)
On a side note: the attack bonus (from the hero or siege) is applied directly on each type; in other words, it does not affect the mounted/unmounted ratio (be your hero on foot, on horseback or flying on Drogon)
