South Downs Way

Perfectly possible on a hybrid/CX with some decent tyres - done exactly that.  Next?

It is possible to do the SDW on a hybrid/CX/Gravel/Enduro bike. You don't need suspension. You just need the wider tyres found on hybrid bikes and the like. I know of people who have done it on 20 year old hybrids from Halfords - it's just about how carefully you ride. The main issue for your young charges would be the nature of the terrain - there is very little flat or rolling ground. For the most part you are either going up or going down. As mentioned, the Downs Link is a fabulous trail for young families.

Friston Forest as well a bit closer to Eastbourne. And the cycle route out of Brighton along the front and under the cliffs towards Newhaven is pretty good too

the sdw is lovely to ride but id do it on a hardtail. there are plenty of very steep sections and there are many many sharp flints for shredding of tyres. when its wet the chalk turns into a frictionless paste which can make for challenging riding and non existant braking. cuckoo trail, the family rides at bedgebury or the downslink would be my choices

I personally don't think cx bikes (particularly the tyres) are tough enough for the South Downs Way.  A friend entered a Wiggle ride that supposedly was perfect for cx bikes (mix of roads, lanes and some SDW) and had to abandon on the SDW section after shredding his tyres.  Possibly he could have been ok with tougher tyres, not sure what he had at the time.  This was around Bury/Cocking area.  The terrain is a mix of grassy paths, proper bumpy fields where the cows have been standing around in the rain, really bumpy/flinty sections, wooded sections where you also have tree routes to cope with...you need some suspension really, especially if you're out for a couple of hours.  And if it has been raining, it can be boggy too.

It's also tough, you're never far from a 300m climb and I doubt a 7 year old would like that!  I've been looking for an "easy" way up to the top of the Downs for years, haven't found it yet.

There is the Downs Link, from Shoreham up to Guidlford, which I think is brilliant for younger riders,  only one real hill up near Guidlford, no cars but close to villages and pubs in case you get into difficulties, or need a  refreshment stop.  I think you can take kids round Tilgate Park near Crawley?  This also extends south beyond the M23 and is more suitable for proper mountain biking.

You can also go along the coast east and west from Shoreham and not have much road riding to do.  Cuckmere Haven hires bikes, and it is beautiful there.  There are trails in Stanmer Park just north of Brighton (again, there will be hills) Houghton Forest west of Amberley.  I could go on!

I can vouch that doing the SDW on a CX bike is rather gruelling. Your ride profile will look like a saw blade. We did it at the weekend, leaving from Hassocks near Brighton. Technical descents and big gradients for the gears on my CX bike.

Road bike on this off road trail, probably not. The Cuckoo Trail sounds more fun for rides with children.

I've done parts of it once or twice. Wouldn't do it on a road bike, CX (or something with CX tyres) should be fine (depending on skills levels). 

Worth checking out the Cuckoo Trail (East Sussex, I think), an old railway line. It's flat-ish (so not gruelling) and you get a nice, car-free ride.