Were the trenches of the western front continuous? (ie could you walk from the ocean to Switzerland in them?)

No. It was certainly possible to move for many miles at a time along one battalion front to the next, but there was never a time when it would have been possible to walk from Switzerland to the beach outpost along a continuous front line trench. Due to 'bite-and-hold' attacks the front was always in flux, even if this only involved small local movements rather than huge sweeping advances. In some areas, at some times, the front line would not even be a trench but a line of occupied shell holes. In other areas overlapping strong points, pill boxes or redoubts stood in for a lateral front line (especially later in the war, when the idea of the 'defensive zone' rather than the 'defensive line' developed). In still others the local geography, geology, or a high water table made trench digging difficult or impossible so the line was built above ground rather than dug into it, using sandbags, stone, revetments etc., and might run through the fortified remains of a shattered village.

Were they continuously guarded, or did soldiers tend to populate areas near towns or where the saw the enemy had been stockpiling troops?

The front line was always manned, but often much more lightly than we might imagine. Lessons were learned early in the war that whilst packing the front line with troops gave an advantage against attack, it resulted in very high numbers of casualties when the enemy artillery found its range. As tactics developed, the frontmost section of line would be held as lightly as possible by sentries, Lewis Gun teams and sniper/observer teams, with the bulk of men held in close reserve in the supervision trench, dugouts, support and reserve lines. If an attack was detected these men could be moved forward rapidly to repulse it, and the signal rockets (which would usually be in a rack outside the company HQ dugout in the supervision trench just behind the forward fire trench) would be set off, which was a signal to the attached artillery batteries to fire on their SOS targets (usually just in front of the wire in no-man's-land).

If the enemy was observed to be particularly strong in a sector, then heavier reserves would likely be held behind the front line as insurance against a potential attack, but this would be in hutments or billets rather than stacked in the line, again to avoid mass artillery casualties. They might be brought forward if the enemy was spotted massing for an attack.

Once an offensive broke through and took a few km of land, were new trenches built?

I've written a bit about this subject in these two answers, along with some links to primary sources which give an idea of when and how a trench would be built:

How did armies in World War 1 build trench systems so close to each other?

What happened after a successful attack against an enemy trench in WWI?