[RPG] How exactly does movement on a mount work?

Short simple answer here. No you can't, you can only mount or dismount once per turn. Player's Handbook, page 198:

Mounting and Dismounting

Once during your move, you can mount a creature that is within 5 feet of you or dismount.

The example turn you described is legal concerning RAW. You would be able to break up your movement and attack both targets as long as your mount has enough movement to get you there. As for the opportunity attack, any time you move outside of a creature's reach without taking the Disengage action you would provoke an attack of opportunity. Luckily, in your example you are using a lance which has the Reach tag. This means that unless the creature you are attacking also has a reach weapon or has some other way to extend its melee reach to at least 10 feet you could choose not to provoke an opportunity attack simply by attacking within your reach and staying outside of theirs.

If for some reason avoiding the opponent's reach cannot be done, let's consider the following:

Page 198 of the PHB:

if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.

That being said, the mount can use it's action to take the Disengage action to avoid being attacked,

The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

and you would not provoke an opportunity attack because,

Page 195 of the PHB:

You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

As for the mount's ability to attack...

Although the steed has an unusually high Intelligence of 6, that doesn't make the mount an intelligent creature who can act independently of you, and it can't attack while ridden. Here's a couple links to a few Tweets from Jeremy Crawford, one of the lead game designers:

Not Independent.

The mount summoned by the find steed spell serves the summoner. It isn't an independent creature.

Still has normal mounted combat options

While ridden, the steed follows the normal mounted combat rules (PH, 198). Unridden, it has normal action options.

The normal mounted combat rules state that a mount not acting independently can only use the Dash, Disengage, and Dodge actions as shown in the quote above in bold.

For trampling charge to work, the mount would need to use the Attack action to attack with its hooves, which isn't in the set of the controlled mount's possible actions.

Trampling Charge. If the horse moves at least 20 feet straight toward a creature right before hitting it with a hooves attack, ...

Therefore, the mount cannot use trampling charge.