1. You can do these while not mounted, and the mounted rules don't forbid any of these, so yes, this is possible.
2. While the rogue can take the disengage action, it has no effect. The mount uses its own movement, not yours, and therefore provokes opportunity attacks on its own. While you're on the mount, you do not provoke opportunity attacks since you're not using your movement to leave the reach of your enemies. However, the mount does provoke opportunity attacks, and enemies can choose whether to target you or the mount with the opportunity attack. The mount itself can still use its action to disengage, though.
3. It can. You and the mount are two separate creatures, so attacks can target either of you. Spells with area-of-effect affect both you and the mount.
4. Absolutely. Since you're two separate creatures, there's nothing stopping a monster from attacking you or the mount, and the creature can even choose who to attack when the mount provokes an attack of opportunity.

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.

a controlled mount ... moves as you direct it, and it has only three action options: 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

$$ \text{Travel Pace and Effects} \\ \begin{array}{|c|c|} \hline \textbf{Pace} & \textbf{[...]} & \textbf{Effects} \\ \hline \text{Fast} & \text{[...]} & \text{−5 penalty to passive Wisdom (Perception) scores} \\ \hline \text{Normal} & \text{[...]} & \text{-} \\ \hline \text{Slow} & \text{[...]} & \text{Able to use stealth} \\ \hline \end{array} $$

Since creature's travelling at a Slow pace can use stealth, we look to the rules on Dexterity ability checks where it says:

Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies ... or sneak up on someone without being seen or heard.

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:

When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions, the GM might call for a Wisdom (Animal Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.

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:

$$ \text{Typical Difficulty Classes} \\ \begin{array}{|c|c|} \hline \textbf{Task Difficulty} & \textbf{DC} \\ \hline \text{Very Easy} & \text{5} \\ \hline \text{Easy} & \text{10} \\ \hline \text{Medium} & \text{15} \\ \hline \text {Hard} & \text{20} \\ \hline \text{Very Hard} & \text{25} \\ \hline \text{Nearly Impossible} & \text{30} \\ \hline \end{array} $$

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.

Training

You can spend time between adventures learning a new language or training with a set of tools. Your GM might allow additional training options... The GM determines how long it takes, and whether one or more ability checks are required. The training ... costs 1 gp per day.