No. Any rules conferred by a weapon, be it magical or mundane, apply only to attacks made with that weapon. This can include, but is not limited to, rules unique to a specific type of weapon, special rules that apply to a particular weapon, or special rules that apply to a magic weapon. For example, if a Wizard is armed with the Sword of Swiftness, attacks made with that weapon have the Strikes First special rule, but the Wizard cannot claim that wielding the weapon allows them to cast Assailment spells at Initiative 10.

To give another example, if a model has a special rule unique to its faction that grants additional rules to a hand weapon (the crew of an Orc Boar Chariot have the Choppas special rule, for example), that rule applies only to actual hand weapons carried by riders or chariot crew, not to weapons belonging to mounts or draft animals that 'count as' hand weapons.

No. If it is noted on a weapon's profile that the model may or must make a specific number of attacks with that weapon, normal attacks or otherwise, that's how many attacks it makes with that weapon.

Yes. As noted on page 213, unless stated otherwise, all models are assumed to be equipped with a hand weapon. A model with two has, by definition, two and may choose to fight with just one, foregoing the Extra Attacks (+1) special rule in favour of some other benefit.

During any turn in which it charged or counts as having charged.

No.