Why must my hopes be crushed from the onset?
I don't think he's saying "Eliza can't deal with these things", just that even though you say you are picking up Eliza from the sounds of your post it still sounds like you are hung up more about double. ....I think...or maybe it's just a reference to your jHP comment? Idk hard to tell
Anyways, I'll see if I can throw some basic pointers that can help.
I have no idea how to approach other than j.HP dash j.HP
Air dash into air throw is very strong for eliza since her air grab is large and you can meterlessly convert into a longer combo.
Instead of using jHP all the time, try using jMK and jLK more. jMK covers horizontal space well to keep people fearful, and use jLK when you get in closer and are above them cause it's a fast button that hits decently well below you. Learn confirms off of both.
Remember that Eliza has a very good range on her cLK and her ground throw, and that her cMK is also a slide that can catch people sometimes.
Don't be afraid to use the divekick servant. Using him is mostly safe, he's a tricky overhead, and he can catch people in the air very well. Once you start using him better, he becomes a thing the opponent has to respect and can also act like a defensive wall too, so it's important to learn.
I have no idea what to do under pressure
Eliza isn't the best character for defenses in general so brace yourself.
#1 tip is to read, if you haven't, and understand
this guide to defense in Skullgirls. It will help more than anything here that I write.
After that, just know what options eliza has. Under Pressure, she has her DP, but you should know it's vulnerable to grabs and it's hitbox on the first hit is fairly low so it isn't super reliable. Her Lady of Slaughter isn't a great reversal. Her air super is fairly decent to use defensively so consider that. Her level 3 is also pretty good and fairly invincible, though it does cast 3 meters, so use wisely.
how to deal with a Peacock/Fukua deadset on zoning on the other side of the screen
Using Eliza's cHK is crucial. It's projectile-armored so it should give you a chance to get in, and it's cancelable too. If you get in close, you can cancel into Divekick Servant or DP. And if you get in and they block it, you can cancel into L Osiris Spiral which is safe on block.
Beyond that, you need to remember to use Eliza's air mobility as well. Using Air dash and jMK to pester them with it's huge range, and using Air dash + air throw when you are close by and they try to upback away.
I'm even worse at facing my trouble characters: Parasoul, Big Band, Squigly.
Let's go through some basic tips for these:
Parasoul - Be very aggresive with Air Dash + jMK to keep her in block, close in the distance and then go in with jLK pressure. Parasoul has a hard time dealing with Eliza's large buttons once Eliza gets into range, and once Parasoul starts blocking it's tough for her to deal with the pressure. Don't forget about cHK approach as well if Parasoul starts throwing fireballs.
Big Band - Don't stick in the air TOO much, because BB can use A Train fairly reliably to catch obvious air approaches that Eliza does. Use cHK approaches to scare BB away from using armor punches, be careful of BB Cymbal ranges, and when you get in close to cymbal range, watch for BB's to jump cause you can usually react with an air dash + air throw to them jumping and catch them using cymbals.
Squigly - Throw out Divekick servants a lot, cause squigly has large hurtboxes and has to respect that. If Squigly jHP is bothering you from afar, try using Eliza jLP since it's disjointed and can hit squigly out of her move. Try and catch squigly's using jLK by getting into Divekick Servant range and using that.
I hope at least something helped.