Given that you appear to have all the rules already, it seems that the problem is in interpretation. Given that, my view is :

1. Should the mounted PC be treated as a normal mount for a PC, losing his turn and becoming subordinate to the rider?

   • Answer: No. The mount is the same level and equal in power

2. If 1 is "no", do they share initiative counts?

   • Answer: No. They each roll initiative, and act on their turns. If one wants to move and the other doesn't, then the rider either dismounts (if the rider is the source of the move) or falls prone (chosing not to move with the mount). As a generous DM I'd allow an acrobatics check to avoid falling prone.

3. If 1 is "no", can the rider use his move action to make the mounted PC move, or force the mount to act by spending his own actions in any other way described by the Mounted Combat rules?

   • As above; if one wants to move an the other doesn't, then it's either a dismount or a fall-prone for the rider, depending on who's turn it is.

   • Per the comments; the mount should only be able to be move normally once per round - whichever commands the mount uses their move action to move, the other still has a move action but can't use it to move (unless it's the rider dismounting). Clearly other sources of movement (e.g. warlords) would still have the expected result - and the mounted combat rules cover that sort of thing already.

4. Can the mounted PC use the rider's base skill check bonuses as described in the Mounted Combat feat (e.g. a human with a pixie on his shoulder uses the pixie's Stealth modifier)?

   • If the rider has mounted combat and the mount is willing, then I would go along with this.

5. How would feats like Holy Steed, which gives a rider's mount bonuses to defenses and damage, be applied?

   • I would go along with this. As a DM, I'd much rather my players bunch up than spread out. A small bonus is worth the penalty.

This answer is wrong. Ignore it, unless it makes your games more fun, in which case, ignore all others.

Yes, you can, and with even more versatility than you thought.

Unlike previous editions or some other games, movement is not a type of action in D&D 5e. It is a resource that you use, almost like a currency that you spend. At the beginning of your turn, you have movement to use equal to your speed. Think of this as a deposit into your account. You can use (spend) movement at any time during your turn, in any amount you have left, any number of times. In between using (spending) movement, you can do anything that your character is capable of, including making attacks, using bonus actions, etc. If you don't have any movement left (account balance of 0), you can't move any more. After the end of your turn, any remaining movement is gone (the account is zeroed).

Using the Dash action increases the amount of movement available that turn by an amount equal to your speed. It's as if another deposit has been made. Therefore, you don't use your Dash to move, you just are able to move more because you Dashed. You can do this at the beginning of your turn, after using some of your movement, or after using all of it.

Now, how does this apply to mounts? Your mount gets a deposit equal to its speed at the beginning of your turn, and you get to direct it how, where, and when to spend that movement, as if it were your own. You can also direct the mount to use the Dash action (no action required on your part), which makes an additional deposit into its movement account. As long as the mount's movement account has movement in it, you can direct it to move as you wish, as many times as you wish, doing anything that your character is capable of in between each movement.

Note: this answer has been contradicted by a Crawford tweet since written.