The mount and the rider each have their own full turn with movement and action. When a player mounts an animal and decides to control it:

The initiative of a controlled mount changes to match yours when you mount it.

The mount's position in the turn order changes but it keeps its own turn on which:

It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

Therefore the mount can move its full speed and finish its turn. After that the character takes his own turn on which it can dismount using half of the speed and move for the remaining half. What the character cannot do though is to mount, ride, and dismount as it would require two character's turns.

There is one thing that requires clarification though. If I understand correctly, ties in initiative are resolved once and for all, which means that after mounting the player has to decide whether it is the mount or the character who acts first during the round.

The only thing I see that may hurt my reasoning is this sentence (emphasis is mine)

A controlled mount can move and act even on the turn that you mount it.

It makes things arguable, but for me everything else is solid enough to consider that there is a mistake and it should be “round” instead of “turn” in this sentence.

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.