Do I need to run electrical wires exiting the breaker box on the exterior wall of the house & traveling across the wall in conduit or can the wires be stapled to the wood siding with steel staples?
In the US -- the general rule is that outdoor rated cables (MC, SE, and sunlight-resistant UF are the only ones you'll see in residential work -- AC isn't allowed in damp/wet locations, nor is NM of course) can be run in the open provided they cannot be subject to physical damage -- if your cable is running anywhere where people can run into it, mow it, whack it, or trip on it, you'll need to install your wire (THWN) or cable (there are no Code objections to pulling intact SE or UF through conduit, although it may annoy the next electrician who works on it) in RMC, IMC, schedule 80 PVC, or RTRC-XW to provide protection from said physical damage, as per NEC 300.5(D)(4):
(4) Enclosure or Raceway Damage. Where the enclosure or raceway is subject to physical damage, the conductors shall be installed in rigid metal conduit, intermediate metal conduit, RTRC-XW, Schedule 80 PVC conduit, or equivalent.
It depends. As @bib implies in a comment, it depends where you are. It depends what type of wire you are using - it must be exterior rated for both water and sunlight exposure. It may depend what type of structure it's on (many things that are fine in the US on a "residence" are strictly forbidden in "commercial work.") In some cases the residential work should at least consider meeting the higher standard as a minor investment resulting in greater safety. Code is a minimum, it's fine to exceed it. In some cases the conduit approach is actually cheaper than exterior cables*, and it's certainly more damage-resistant.
*Any wire in a conduit that is outside must be rated for wet locations. However, it's not uncommon for 3-4 wet-rated wires to be cheaper than the same number and size of wires grouped as an exterior cable to an extent that will pay for the conduit to put them in.
They need to be in conduit for 2 reasons. First, exposed wiring needs to be protected from damage and stapling bare wire to the siding would not qualify. Burying it at least 24" would qualify, but any portion above that would still need to have some sort of damage protection (conduit). Second, exterior boxes, receptacles, and entry/exit points need to be weatherproof. Conduit provide a means to weatherproof the ends of the runs, and I doubt any other method would be suitable for a horizontal run.