These rules change significantly if the mount isn't war-trained or if making ranged attacks instead of melee attacks. I figure as a paladin 12/wild plains outrider 3, you're probably mainly concerned with melee attacks while on a war-trained mount.
Both mount and rider can make full attacks in melee on the same initiative count
The opponent's gotta be right up in the rider and mount's collective grill and the pair won't be able to move very far during that round, but if that's okay then both man and beast can take the full attack action and pound on nearby monsters.
For both mount and rider to make full attacks on their initiative count...
- the war-trained mount can't have moved more than 5 ft. and
- the rider must take a free action to make a Ride skill check (DC 10) to fight with warhorse.
First, according to Mounted Combat on Combat while Mounted,
If your mount moves more than 5 feet, you can only make a single melee attack. (PH 157)
So for both mount and rider to make full attacks, mount and rider are limited to the mount's 5-ft. step.
Second, according to the skill Ride on Fight with Warhorse,
If you direct your war-trained mount to attack in battle, you can still make your own attack or attacks normally. This usage is a free action. (PH 80)
Emphasis mine. Further, the Monster Manual entries for the heavy and light warhorse share this text: Either creature
can fight while carrying a rider, but the rider cannot also attack unless he or she succeeds on a Ride check. (274)
Although it's a bold assumption, for the sake of playability I think the Ride skill check mentioned in the light warhorse and heavy warhorse entries is the same Ride skill check made to Fight with Warhorse, therefore a successful Ride skill check (DC 10) permits full attacks from both mount and rider. Note that failing the Ride skill check to fight with warhorse probably means the rider cannot attack, but ask the DM.
The rider cannot make a full attack in melee after his mount makes a charge
Most creatures must take a full-round action to make a charge and, to do so, most creatures must move at least 10 ft. As previously mentioned, a mount that moves more than 5 ft. limits the rider to a single melee attack.1
The prestige class wild plains outrider (Complete Adventurer 92-4) gains at level 3 the extraordinary ability wild plains offensive, which says
Starting at 3rd level, a wild plains outrider can make a full attack with a melee weapon as long as his mount takes only a single move. (93)
This ability lets the wild plains outrider circumvent the lone attack clause that usually occurs after the mount's moved more than 5 ft., but this in no way changes charge, which almost always takes a full-round action, even if the foe is but 10 ft. away. For example, a level 3 wild plains outrider can direct his mount to take a move action to move up to its speed; take a free action to make a Ride check (DC 10) and, if successful, direct his mount to make a standard attack; then take a full action to make a full attack in melee. For most folks, that last part would have only been one attack.
1 This DM would allow the rider can still take the full attack action, but nonetheless limit the rider to a lone attack. Maybe the rider took the feat Circle Kick (SF 5) or Dirty Fighting (Dragon #303 28). Kidding! Nobody takes Circle Kick or Dirty Fighting.
There are not rules for riding with a non-military saddle vs riding bareback, though that would be within the rights of the DM to rule as such (the rules are currently silent on the difference, and only provide an explanation that a military saddle provides advantage on checks to stay mounted).
Dash is one of the normal action types for a nonintelligent mount, so I would definitely not make you make a check in that case unless another circumstance arose.
There are effects that would dismount you. The most common is an effect that moves your mount against it's will (DC 10 Dex save).
The rules for mounts are outlined in two places p198 and p155. It would be good to review these two pages to get the full picture on what is and is not codified currently about mounts.