I have a question regarding these 2 cards. Can I play Solemn Judgement then *Barrel behind the door* to switch the cost of 50% life points to my opponent? Let me know thank you.

This is the wrong section to ask a question of that nature; however, I'll still answer it.
Barrel only works on effect damage, not costs. The paying of half of your LP for the activation of Solemn is a cost, and therefore cannot be transferred to your opponent by Barrel.

ah my bad thanks for verifying it for me:)

And it also only works on "damage".

If a card, as few as they may be, simply "decreases" your Life Points or makes you "lose" Life Points or anything besides "damage" specifically, then Barrel Behind the Door cannot be chained then either.

You can also only activate it if the effect you want to chain to actually creates a Chain Link, so you cannot chain Barrel Behind the Door to Continuous Effects.

If the damage is not being done until the card is resolving, you can't chain Barrel Behind the Door (like to Graverobber if you decide to use the Magic Card you take by its effect or Power Bond during the End Phase).

The damage must also be "guaranteed" damage. If the effect involves dice rolling or coin flipping, you can't use Barrel Behind the Door.