He must escape/break-free from each separately

However, there are only 2 restraints present: the manacles and the rope - the fact that you used 2 lengths (or 7 or 15) is irrelevant - you tied him up once.

He only gets one shot to escape and one shot to break each

Your DM was unfair; he was inducing Rolling to failure. The NPC can only attempt to escape from each binding once and only attempt to break each binding once. If he fails he is not agile/strong enough to do it trying again won't make him so unless circumstances change.

Allowing the NPC to do this means that he will succeed eventually (always assuming that it is possible for him to do so) thanks to how "rolling to failure" works out mechanically.

How I would rule it

The manacles have a DC20 Strength check to break and a DC20 Dexterity check to escape. The rope has a DC17 Strength check to break and a are an opposed check between his Dexterity and your Dexterity (Slight of Hand) [Sage Advice]. If you were wise, you would get someone to Help you tie him up to get Advantage.

Being tied up, I would give him Disadvantage on all of these because he is under the restrained condition. (See Appendix A).

So that's 2 chances to get out of each and he has to get out of both to escape.

No, it can't lift a party across a river

It can create a single rope "bridge" that the party would then need to climb/walk on to cross the river.

Holding vs Lifting

The Rope of Climbing (DMG, 197) does not Lift anything per the description. What it does is:

If you hold one end of the rope...the rope animates. As a Bonus Action, you can Command the other end to move toward a destination you choose.

This is clear in that you are holding one end of the rope while the other moves towards the destination of your choice. Once the end arrives at it's final destination, you (and others) would then still have to climb the rope. It does allow for easier climbing to limit risk by:

If you tell the rope to knot, large knots appear at 1-foot intervals along the rope. While knotted, the rope shortens to a 50-foot length and grants advantage on checks made to climb it.

It is not meant to carry you up. It simply removes the requirements of using a grappling hook and still requires someone to climb it. If it was meant to carry you up, there would be no need for the climbing mechanics or the knotting to assist in climbing.

As for horizontal/arc movement, that's perfectly acceptable as well. There are no stated limitations on how the rope moves 10' and in fact does state:

move toward a destination you choose

This allows the rope to move in any direction you choose.

No language on what happens if you overload

Frustratingly, there is no language on the results of putting more weight on than the rope can handle. Other magic items do provide such effects, but it is not described in the DMG and is therefore under the purview of the DM. But given that it doesn't say the Rope is destroyed, it's more likely that it just can't continue to lift/move if overloaded.

Get creative

There is language in the description that just begs for creative thinking (emphasis mine):

You can also tell the rope to fasten itself securely to an object or to unfasten itself...

You can not only use it like a grappling hook that'll always find it's mark, but it can be a 60' long extension of your hands as well. Need those keys? Rope gets 'em. Gotta pull that lever across the chasm? Send the rope! It is not just a means of getting from Point A to Point B, but also a means to bring/interact with things at Point B.