Slate work on the older stone and Victorian property the North East is full of: full slate roofs, patch repairs, matched replacements, valleys and hips. Slate is unforgiving because every course shows, so gauge, headlap and a straight line matter as much as watertightness. Reclaimed and new, matched to what is already on the elevation.

New or reclaimed slate matched for size, colour and thickness, so a repair blends into the roof rather than patching it in a different shade.
Set out properly before the first course goes on. Get the gauge wrong and the roof both looks wrong and lets water past in driving rain.
Cut work, lead valleys and hip detailing done by hand, which is where a slate roof is either right or obviously not.
Older houses have settled and nothing is square. The slate is set out to the roof that is actually there, not to a drawing of it.
Usually, yes. Slate varies in size, thickness and colour depending on where it was quarried and when it went on, so matching is a case of getting up, measuring what is there and sourcing new or reclaimed to suit. On a front elevation that matters a great deal, because a patch in the wrong size or shade is visible from the pavement for the next forty years. On a rear elevation you have more latitude and it is worth saying so rather than spending your money on reclaimed you will never see.
Good reclaimed slate is excellent and often a better match on a period roof. The catch is that reclaimed is only as good as the sorting, because slate that has already spent a century on a roof can be delaminating or have nail holes in the wrong place. Callum will tell you honestly whether reclaimed is the right call for your job or whether new slate, which comes with a known thickness and a known life, is the better spend.
Almost always nail sickness. The slates are fine, but the nails holding them have corroded through, so they let go one at a time, usually after wind. On a roof where that has started, the rest of the nails are the same age and the same story, so refixing them individually turns into an annual appointment. That is the point where it is worth costing a re-slate against another five years of repairs, and getting an honest opinion on which makes sense.
New slating is nailed, which is how a slate roof is meant to be fixed. Clips, or tingles, are for putting an individual slipped slate back into an otherwise sound roof without disturbing the courses around it, and used that way they are perfectly sound. Where clips get a bad name is when a whole roof is held together with them, which is a sign somebody has been repairing a roof that needed replacing.
Only within limits, and the limits are real rather than fussy. Slate relies on gravity and lap to shed water, so below a certain pitch it will let driving rain past no matter how well it is laid, and the answer is a bigger headlap, a different slate size, or a different covering entirely. If your roof is at the margin you will be told, because a slate roof laid too flat is a leak with a date on it.
Yes, and the main thing it changes is what you are allowed to put back. In conservation areas and on listed property there is usually an expectation, and sometimes a condition, that the roof is re-covered in matching natural slate rather than a substitute. Worth checking with the council before ordering, and worth allowing for in the budget, because the matching material is generally the dearer option.