Yes, A monkey riding a dog can make attacks against whomever they choose
It has taken me a while to piece together all the rules supporting this, and make the scenario plausible. This is what I have came up with:
Abilities - 13 Strength (Mount)
Skills - Atleast 1 point in Ride.
Feats - Ride-By Attack (Monkey), Mounted Combat (Monkey), Power Attack (Dog), Improved Overrun (Dog)
Ride-By Attack - "While mounted and charging, you can move, strike at a foe, and then continue moving." The feat goes on to state that "You and your mount do not provoke an attack of opportunity from the opponent that you attack". This will allow your monkey the opportunity to attack their foe somewhere along the mounts movement course.
Improved Overrun - "You do not provoke an attack of opportunity when performing an overrun combat maneuver and targets of your overrun attempt may not choose to avoid you.". Without this, your opponent can simply avoid you and you will never have the opportunity to strike at them, as well as take provoked attacks of opportunity.
Overrun - "During your movement you can attempt to move through a square occupied by an opponent... This is a standard action taken during your move or as part of a charge." In this case you would be charging. "You can only overrun an opponent who is no more than one size category larger than you"
For simplicity, you assume that you share your mount's space during combat (pg 202 CR). This means that if you are a medium creature riding on a large horse, you are treated as a large model in combat. If you are a tiny creature (One who would normally take up 1/4th a square) riding on a small dog, you are treated as a small model in combat. Note: The rider does not get the extended range of the mount.
How Combat Works
Because you are not allowed to move through an opponent occupied square unless that opponent is helpless and because you cannot end your turn in an opponent occupied square you must perform an Overrun maneuver. When you attempt the overrun maneuver you must have the Improved Overrun feat, else the opponent can simply avoid you.
You start by making a combat maneuver check. If your attack fails, then you stop in the space directly in front of your opponent, or the nearest open space in front of the creature, you do not get an attack. If your attack exceeds your opponents CMB you may move through your enemies space and then make your run by attack. If your attack exceeds your opponents CMB by 5 or more, then your opponent is knocked prone and you get your attack against a prone opponent.
Here is the tricky part! Because you can only overrun an opponent no more than one size category larger than you, your dog must be the one doing the overrunning... This works out because it is unlikely that your monkey is going to have 13 strength for the overrun feats, it also makes it difficult because the mount would have to be treated as the Druids Animal Companion (Paladins and Cavalier mounts are treated this way), and have at least 3HD to select feats requiring BAB+1.
MrLemon addressed the reach weapon portion of your question.