Do weapons with this description provide a free trip attempt after a successful attack (Dealing Damage), or is one just using the weapon instead of the unarmed attack hitting the touch AC, dealing no damage?
The latter—you can use the weapon to perform a trip attack, but the only differences from performing it unarmed and with such a weapon are those explicitly listed in the section following your first quote, about tripping with a weapon:
Tripping with a Weapon: Some weapons can be used to make trip attacks. In this case, you make a melee touch attack with the weapon instead of an unarmed melee touch attack, and you don’t provoke an attack of opportunity.
If you are tripped during your own trip attempt, you can drop the weapon to avoid being tripped.
(Trip special attack description, Player’s Handbook pg. 159)
So the only differences are that you don’t provoke an attack of opportunity, and you don’t risk being counter-tripped. Well, and the fact that you are using the weapon, which may matter if, for example, the weapon has reach, or provides a bonus on trip attempts, or otherwise has properties that apply to melee touch attacks and/or trip attacks. But for (most?) weapons, damage isn’t one of those things that applies to those things.
Note, however, that there are ways to get free trips after a successful attack. This kind of thing is even the signature property of many canine monsters (including dogs and wolves):
Trip (Ex): A wolf that hits with a bite attack can attempt to trip the opponent (+1 check modifier) as a free action without making a touch attack or provoking an attack of opportunity. If the attempt fails, the opponent cannot react to trip the wolf.
(Wolf monster description, Monster Manual pg. 283)
A player character can gain this ability through certain forms of shape-changing magic to become such a creature, or with the Knockdown feat (which works the same but requires that you deal a minimum of 10 damage with the attack).
If the above answer is yes, a free trip attempt after dealing damage, then; if one has the Improved Trip Feat, do they still get the free attack after a successful trip?
The answer is “no,” but since there are other ways to achieve that, this is still a relevant question—and the answer here is “yes.” Creatures with such free trip attacks still benefit from Improved Trip’s extra follow-up attack, so an attack that leads to a trip can then cause a follow-up attack.
Note that you cannot trip a creature that is already prone, and if you do not trip a creature, you do not trigger Improved Trip’s follow-up attack. Therefore, at least against a single target, you only get one extra attack this way.
It’s also worth noting that attacks of opportunity happen before the thing they interrupt—which means if you take an attack of opportunity against someone trying to stand, you are attacking them while they are still prone and cannot be tripped. So you get your attack, but don’t get to trip them, can’t trigger a follow-up Improved Trip attack, and they get to stand up afterwards and then act however they like while standing. (If they provoke another attack of opportunity after they’ve stood, and you can still make an attack of opportunity, e.g. because of Combat Reflexes, then you can attack, trip, and attack again. This is the basis for “trip-lockdown” builds, and it’s fairly effective as far as it goes.)