Will our party have enough healing (paladin and warlord) to get by?
Since the warlord is a primary leader who can heal right from the start and the paladin could add some reasonable healing by himself, I'd say your party is ready to go. If you realize that healing is running thin in combat, perhaps your party should consider some of the leader multi-class feats for additional minor-action healing surges or magic items that add to healing surge value or allow [additional] healing by themselves.
Some suggestions for magic items:
-  Amulet of False Life[DDI] (daily item power, minor action, gain temp hitpoints equal to healing surge value when bloodied)
-  Amulet of Vigor[DDI] (daily item power, free action, if you spend a healing surge regain additional hit points equal to healing surge value)
-  Amulet of Life[DDI] (encounter item power, free action, when spending a healing surge spend additional healing surge)
-  Diamond Cincture[DDI] (at-will item power, minor action, use diamond and spend healing surge, all diamonds recharges after extended rest)
If this doesn't help much, perhaps a little retraining on the warlord's and paladin's part could help (foregoing powers with more damage in favor of powers that allow extra healing).
What strategies (tactical or otherwise) are there to reduce the amount of healing needed?
Damage mitigation and enemy lock-down are king in a group with stretched or limited healing. Anything that allows you to prevent the enemy from retaliating at all or greatly reduces his effectiveness is wonderful, especially certain combos of status effects - e.g. "prone + dazed" or "slowed + blinded" since both eat up an enemy's actions and reduce his ability to hurt the party.
Powers that provide resistances or regeneration are a great help, too, since they reduce the necessity of actual healing. Temporary hitpoints are equally handy. For example, a single lowly Protective Roots[DDI] can prevent several hundred points of damage on the whole party in a single combat.
Combat tactics in such a situation often include lots of damaging/controlling zones and forced movement, (ab)using the ability to damage/control enemies multiple times with a single effect.
How do I teach my party members to not be so reckless and dependent on healing?
If the players won't listen to your out-game words they hopefully have learned when rolling up new characters. I'm dead serious about this. If a player doesn't change his tactics and soaks up all the group's healing abilities by trying to please his ego he should learn the hard way that the party's (cleric | warlord | bard | leader-of-choice) is not his personal nurse.
It's all about fair play. Nobody wants to play the darn heal-bot for the guy doing all the fun stuff in battle, so it should be a matter of politeness for all players to act accordingly so that all players can do interesting things in combat and not just waste actions/magic items/powers known to keep the self-centric melee-guy alive.
A fighter or barbarian or ranger ignoring the party's healing limits is being egoistic in multiple ways: not only does he soak up all of the party's healing resources and thus prevents other players from doing cool things, but he also increases the risk of a TPK by orders of magnitude.
If the characters don't listen then there's a simple way of teaching them: letting them hit negative hitpoints a few times (of course without actually dying) may get the point across that there isn't much healing around currently. Especially if the character has never before really been injured (or there was always a healer around to patch him up right after) it may be a shock to discover that there was now some actual risk of death.