Persistent spell simply allows a spell to invoke two saving throws instead of one. If the target fails either of the saving throws, they suffer the full effects of the spell unless they have the ability Evasion or Improved Evasion (in the instance of spells requiring Reflex saves)
In the instance of Constricting coils, a Will save is triggered and if failed the target is subject to Paralyzation and constriction. Since the spell is persistent, whenever a will save is attempted (at the beginning of that round) two saves must be rolled instead to determine if the target can break free of the paralysis/constriction.
In the instance of Black Tentacles, that spell does not have a saving throw, therefore it is not subject to being Persistent. At the bottom of the Persistent spell metamagic feat, it reads:
Spells that do not require a saving throw to resist or lessen the spell’s effect do not benefit from this feat.
Clever, but unlikely
In order for Catapult to work, it requires:
one object weighing 1 to 5 pounds within range that isn’t being worn or carried.
This is the requirement that Delayed Blast Fireball must meet. The question is: Does it?
Delayed Blast Fireball isn't explicit in the definition, but it does state:
A beam of yellow light flashes from your pointing finger, then condenses to linger at a chosen point within range as a glowing bead for the duration.
How much does a bead of light weigh?
In order for that bead to qualify for Catapult, it must be at least one pound and less than five pounds. As Dale M states, that's likely a DM call. But how should the DM approach?
Well, it's a beam of light that condenses into a bead. How much does a bead of light weigh? I'd think that a bead of magic light does not weigh anything, but another DM may believe otherwise. Then it's a question of does a bead of light weighs more than 1lb. Even if it did have a mass, it's unlikely to weigh more than 1lb.
Without meeting the weight requirement, you can't use Catapult to launch the Bead for extra damage (pending the save failure).
Throwing the Bead
There is another option for interacting with the Delayed Blast Fireball:
If the glowing bead is touched before the interval has expired, the creature touching it must make a Dexterity saving throw. On a failed save, the spell ends immediately, causing the bead to erupt in flame. On a successful save, the creature can throw the bead up to 40 feet.
This does imply that it can be handled, but it doesn't answer weight or anything else. Ultimately, this is a magic bead made of condensed light and it does not necessarily mean it has mass. It just means that you can throw the magic bead. If a DM does rule that it has mass, then it still needs to weigh more than 1 lb (see discussion above on weight of condensed light.)
Best Answer
You can't carry it around or throw it in a glass jar
The bead stays in one place for its entire duration
Delayed blast fireball says:
The spell tells us that the bead will stay at the given point for the entire duration at the spell. It does not outline any way the bead can be moved except by throwing it.
Remember, the spell targets a point in space and not an object. Targeting a point in space touching the jar doesn't mean the bead will follow the jar.
In the end, putting it in a jar isn't going to give you the ability to move a spell effect that says that it stays put.
You can throw it, but you have to touch it and take the risk
Putting it in a glass container will not allow you to dodge the risk that comes with the throwing clause in the spell. To throw it, a creature must touch it which then triggers the possibility that it ignites right there. If it is in a jar, you are not touching it, therefore you cannot throw it.
You can only extend the duration of the spell once
The rules for metamagic say:
Extend does not provide an exception to this rule, so you can only apply it once to your casting of delayed blast fireball (see this Q&A for more details). That means the duration will be extended to 2 minutes for a hefty 32d6 of damage.