No, because the mount acts on a separate turn from yours
When you use your action to Dash, you can use a bonus action to make
one melee weapon attack or to shove a creature.
- If you move at least 10 feet in a straight line immediately before taking this bonus action, you either gain a +5 bonus to the attack's damage roll (if you chose to make a melee attack and hit) or push the
target up to 10 feet away from you (if you chose to shove and you
So the basic idea of your step 1 is perfectly fine. You take the dash action, which is allowed and this means that your movement increases. However, do note that this doesn't affect your mount at all (so no "increasing speed by 10 ft" for example). Also it is good to note here that dash does not modify any speed at all; it simply grants additional movement.
The problems start around step 2 however.
Since you have used the dash action, you may now take a bonus action attack against any enemy in range. However, in order to get the +5 bonus damage as well you must first move at least 10 feet in a straight line immediately before taking this bonus attack.
You might think that moving your mount would work, but it does not. This is because a controlled mount seems according to the rules to act on a turn separate from yours. Per Jeremy Crawford:
A rider and a controlled mount have separate turns, but they have the same initiative, which means you decide which one goes first.
Thus, you cannot intersperse your and its movement and actions (see this answer for more details on this and the debate surrounding it).
This means that you must end your turn in order to have the mount move towards the opponent. Since it is no longer your turn, you cannot take the bonus attack (bonus actions must be taken on your turn) or the +5 damage (because it is no longer immediately after your dash action).
However, if you dashed then dismounted and moved 10+ feet in a straight line towards an enemy you would definitely be able to get your bonus attack with +5 damage.