By RAW, yes to both.
When an enemy moves out of your reach, it provokes an opportunity attack from you, so when an enemy leaves the reach of your dagger or your whip it provokes an opportunity attack from you (rules on Combat):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
The Reach property clarifies that such an opportunity attack can be provoked by a reach weapon based on that weapon's reach (rules on Equipment properties:
Reach. This weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it.
If you are wielding both a dagger and a whip, the above means an enemy provokes an opportunity attack when it leaves the reach of either weapon, which will occur if an enemy moves from 5 feet to 10 feet away or from 10 feet to 15 feet away. Note that the wording of the Reach property clarifies that your extended reach for your whip does not somehow extend your reach for the dagger. Your opportunity attacks with the whip have reach but your opportunity attacks with the dagger are still adjacent only. This won't be relevant to our scenario, for a reason you will see below.
If you have the War Caster feat, when an enemy provokes an opportunity attack from you due to movement, you can forego the opportunity attack and instead use your reaction to cast an eligible spell (rules on War Caster):
When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack.
There are two very significant rules our scenario hinges on:
By RAW, the only requirement to trigger the War Caster option is that a hostile creature provokes an opportunity attack with its movement. There's no rule that you have to actually make or be willing to make or be in the process of making said opportunity attack with a certain weapon; in fact, you're specifically skipping that opportunity attack. There's also no restriction on the nature of how that movement provoked an opportunity attack, only that it did.
By RAW, when you use the War Caster option, you're not getting an opportunity attack with the spell. You are skipping the opportunity attack and instead using your reaction to take the special spellcasting option afforded to you by the feat. So the question is not whether you "get an opportunity attack with a spell" but whether you "get to cast a spell in lieu of an opportunity attack."
Then we have enough to solve each of the cases in our scenario.
Can you apply War Caster when a creature leaves the reach of your dagger (moves further away than 5 feet)? It provoked an opportunity attack from you by movement, right? Then yes, cast that spell.
Can you apply War Caster when a creature leaves the reach of your whip (moves further away than 10 feet)? It provoked an opportunity attack from you by movement, right? Then yes, cast that spell.
In summary, War Caster's trigger and thus its applicability is the same for the dagger and the whip (except for the distance at which it triggers): when an enemy provokes an opportunity attack by moving. There's no difference between the weapons with respect to this trigger.
This RAW answer does not rely on or consider any designer clarifications, since those aren't RAW. It does account for the current version of the text (meaning the errata available as of this post).