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.
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.
Best Answer
There are two systems that govern stealth: the Hide action and Travel Pace
Hiding
If you are scouting a specific enemy, you should use the Hide mechanics which involve using the Hide action (to provide a DC for any potential onlookers). A controlled mount, such as your horse, cannot use the Hide action since the only actions it can take are Dash, Disengage, and Dodge.
If you are not mounting the horse at the time, there is no reason you couldn't try to convince it to use the Hide action, see below for some idea how the Animal Handling would work. This could be made easier with spells such as beast bond or speak with animals.
Travel Pace
If you are simply scouting ahead of your party while travelling in an attempt to be prepared to warn your party of threats ahead of time, use the Travel Pace rules
Since creature's travelling at a Slow pace can use stealth, we look to the rules on Dexterity ability checks where it says:
As such, all creatures involved in the scouting (albeit at a Slow travel pace unless your mount is your Ranger's Companion) could use a Dexterity (Stealth) check to avoid notice. This means your party would also have to travel at this pace or they would catch up to you fairly quickly. To convince your horse to move stealthily would involve this section on Animal Handling:
Animal Handling
Nothing specifically allows for the instruction of your mount to move stealthily but if adequately trained, it would likely be achievable (I would probably consider exotic mounts proficient in stealth such as the tiger to already be somwhat "trained" in this way). For a creature untrained, it is likely almost impossible. Your GM could assign a relevant DC for Animal Handling with these rules, basing the difficulty on the level of training for stealth the mount has received:
I would rule that a mount with no training would result in a "Nearly Impossible" DC and move up the table for each level of training or comparable experience. I would use the Downtime rules for Training as a guide (see below, emphasis mine). Perhaps with each series of training requiring 3 successful Animal Handling checks before 3 failures (checks made periodically with 1/5 the total time and then are no longer needed after 3 successes, DC based on the level it has before training), you improve the mount's understanding of particular situations for stealthy movement. I would allow proficiency in stealth to give the mount a starting level of Medium. Once it is trained beyond Easy, I would stop requiring checks for this activity.