No. You use the rust monster's Rust(Su) ability as a standard action in the form of a melee touch attack. When you make the combat maneuver, you replace your attack with the combat maneuver.
If I make a trip attempt with a rust monster's antennae, and my foe is in a heavy armor that covers their feet, do I get the benefit of the rust ability as well as the trip?
No, the rust monsters Rust(SU) only applies to the antennae attack. When you make an attack roll, you may "substitute" it for a combat maneuver. The only time you specify what you are performing the combat maneuver with is in scenarios like disarm, where wielding a weapon grants you a bonus / not wielding a weapon allows you to take the weapon if you succeed at a penalty to the maneuver.
Can a rust monster deliver a touch attack with their antennae and get the rust ability on top of the touch spell? (Even if the spell is rusting grasp?)
Yes. You would use the antennae combat bonus and risk losing the spell like you would under normal circumstances.
Rather than sundering with the antennae, could I attempt a disarm and still rust the weapon if successful, in addition to flinging it to the ground?
No, for the same reason as the first question. If the rust monster was capable of performing combat maneuvers with it's antennae then it would have been mentioned in the monsters stat block. If you continue reading I go into more explanation about SU abilities and standard actions, if the Rust(su) ability specifically stated that it was not a standard action to perform then there would be a possibility.
Lets start with the standard actions: Supernatural Abilities (su) is usually a standard action unless otherwise defined by the ability's description. Under the Rust(SU) description found under the rust monster character sheet: http://www.d20pfsrd.com/bestiary/monster-listings/aberrations/rust-monster it says it is a touch attack and it does not mention it being any other form of action = it is a standard action to perform. You cannot perform two standard actions normally in a round so in order for you to trip (a standard action) and use your Rust (su) (with your antennae : another standard action) you would have to use a full-round action and replace one of your attacks with a trip and one with the Rust (su).
Any combination of a creature's attacks during a melle full attack can be replaced by a trip, disarm, or sunder maneuver. (any maneuver that says "in pace of a melee attack"). when doing this, the calculation for the creature's CMB uses base attack bonus of the attack that was exchanged for a combat maneuver.
If we continue to the trip section you will see a note about how making trip attempts do not require a weapon with a "trip" special feature. You can use any weapon for example, a longsword or an unarmed strike, the only advantage for using a weapon with a "trip" special feature is so that if your combat maneuver fails by 10 or more you can drop the trip weapon instead of falling prone yourself.
If you were curious if combat maneuvers did damage they do not: You can find numerous questions and resources online where people ask the same question and in the end: trips are done in place of an attack, the result is knocking your opponent prone not dealing damage normally.
Here is another similar source: Do flaming weapons with the trip special weapon feature deal their extra damage on trip attempts? You are simply substituting the flaming extra damage for the Rust (SU)
If you don't discharge the spell in the round when you cast the spell, you can hold the charge indefinitely. You can continue to make touch attacks round after round. If you touch anything or anyone while holding a charge, even unintentionally, the spell discharges.
The entire section about "Holding the Charge" Was what I was referring to OP