Are gutter guards worth putting on?
Guards can reduce some debris entry, but they still need upkeep and must handle the runoff reaching them. Compare the leaves or needles around your house, how the guard can be cleaned and whether its design suits the gutter. The right choice depends on that combination.
Where should the downspouts go, and where should the water end up?
Placement follows the low end of each run and the corners, with enough outlets that no single length of trough is asked to carry more than it can move. Where the water lands matters as much as where the pipe is. Discharge belongs well away from the foundation and on ground that keeps carrying it away, clear of a planting bed against the wall and clear of any lower roof section, because dropping it on a lower roof just moves the same water down a level and shortens the life of whatever sits under it. Extensions, a splash block or a buried line are all options, and your roofer can lay out which one suits the grade you've got.
Do gutters actually affect the roof itself?
They affect it at the edge, and the edge is where roofs tend to go first. Water that backs up behind a run, or spills over the back of it, soaks the fascia and works into the edge of the decking under the first course of shingles. A missing or misaligned drip edge speeds that up, because the drip edge is what decides whether runoff enters the trough or slides behind it. The middle of a roof can be in fine shape while the last few inches at the eave quietly rot out, and once the deck at the eave has gone soft you're into roof repair rather than gutter work. That's why a roofer looks at the gutter line during an inspection, and at the roof edge during a gutter job.
Can the existing gutters be re-hung instead of replaced?
It may be possible when the gutter is sound and the problem is pitch or attachment. The supporting fascia must be sound, and the gutter must be appropriately sized for its drainage area. The roofer can explain whether adjustment, related trim repair or replacement is the better scope.
The gutters are dented after a hailstorm. Do they have to be replaced?
Not always. Check whether the run still drains, remains securely attached and has sound joints. Damage that is only visual differs from a crease, split or displaced component affecting performance. Have the affected gutter and nearby roofing assessed after a storm.
Do the gutters have to come off when the roof gets torn off?
Not usually. They come off when the fascia behind them has to be replaced, or when the drip edge can't be set correctly with them in place. If yours are coming off, ask whether they're going back up or being replaced, and get that answer onto the estimate before anybody starts.