No, it is not balanced

Warlock sub classes tend to be quite powerful, as a rule. So it's understandable you would want to give your class some major benefits. However, there are several balance problems with this class.

Access to overleveled spells

Two of your four class features above are dedicated to the daily casting of a spell. Given Warlock's limited abilities to cast spells above 5th level, this is a reasonable idea for a class feature. However, in both cases you've given the warlock access to a spell several levels before another caster would get them.

A full caster (e.g. Wizard) would normally have access to 7th level spells at 13th level, and 8th level spells at 15th level. But you've given this class access to the 7th level spell Plane Shift at 10th level, and the 8th level spell Maze at 14th level.

The spells in this game are balanced around the assumption that characters get access to them at the appropriate levels. In fact, your version of Plane Shift is considerably more powerful than the usual version, since you do not require a different 250 gp focus for each plane. So it's likely that these earlier access to superior spells might cause some balance issues in play.

I would recommend allowing the warlock to cast Plane Shift as described as its 14th level feature, and replace the 10th level feature with something else. That might be more balanced compared to other classes.

Resource-free healing

As it is currently worded, your "severance" feature ensures that after any combat, you can quickly heal yourself up to half hit points without spending any resources (other than bonus actions), unless you've lost a limb. This is an extremely powerful ability, that essentially means you will always be at half or higher hit points after every combat.

The ability to regain a limb is also quite powerful. Although many DMs don't include the possibility of limb loss, it is a major impediment to a character if it happens. The lowest cost way to heal it normally is the 7th level spell Regeneration, which usually could only be cast at 13th level. To give a character access to this at 6th level is very likely to cause balance issues.

If the ability to regain 1d4+Cha hit points was limited to once a short rest, or took an action and only gave you temporary hit points, and the ability to reattach a limb was removed until 14th level, this feature might be more balanced.

Your expanded spell list may be overpowered

Haste is one of the best buff spells in the game. The ability to cast it twice per short rest at 5th level may easily be unbalancing on its own. And the ability to cast Death Ward multiple times throughout the day (essentially spending any unspent spell slots on it for another party member whenever you are about to take a short rest) is a serious balance issue on its own. It might be ok to have one of these spells on your expanded spell list (although either would be a campaign defining class feature), but I'd heavily recommend against having them both.

Serenity's Shadow is problematic when combined with other characters

The ability to have an "always on" advantage to stealth checks is quite powerful to begin with. Add to that a +1 or +2 to AC at all levels, and you've got a pretty sizable advantage over some other classes. But these features in and of themselves are unlikely to unbalance your character, and is more or less in line with what other warlock classes get at first level (especially front loaded ones like the Hexblade).

These features may cause some issues when multiclassing (Rogues and Paladins would both jump at the opportunity to have these "always on" abilities). However, that isn't necessarily out of balance with established classes either (again, compared to the Hexblade).

What makes Serenity's Shadow problematic is the ability to inflict vulnerability to "slashing, bludgeoning, and piercing damage until the end of your turn." Compare this to the Grave Cleric's chanel divinity ability: this gives vulnerability to one attack, and it costs the Cleric's action to use. Your ability will work for an unlimited number of attacks, and lets the warlock do a normal attack in order to use it.

As a simple example of how this could be abused, consider a team with a Necromancer Wizard, a Rogue, and your Lady of Pain Warlock. The Wizard could instruct all of its skelletons to use the Ready action and fire at your enemy when you attack it. Similarly, the Rogue could Ready an action to fire with the same trigger. All of these attacks would then deal double damage to the target (since the imposition of vulnerability has no saving throw and does not require the Warlock's attack to hit), essentially giving most of your team a free turn due to your one ability.

This ability wouldn't be game breaking for a lot of parties. After all, at higher levels most martial characters (who are the ones who usually deal the most bludgeoning, piercing, and slashing damage) do more damage on their own turns than double the damage of one attack, so it would be a mistake for them to Ready actions and attack once on your turn. But this ability is potentially encounter ending enough that an entire party might design their characters based entirely around it. I would not be at all surprised to find this happening.

Don't give up! You're off to an intriguing start

A lot of your proposed features are extremely interesting. I especially like how you have them all tied around coherent central themes. But I think most if not all of the features proposed here need some rebalancing before it's a fully balanced Patron.

Real busted past level 5

Let's start from the top.

Expanded Spell List - seems alright, but the spells chosen don't seem to thematically fit a genie...? I was expecting more divinations. It's not overpowered, but just seems kinda weird.

Lesser Wish
Djinn: Decent at low levels, upgrades into Legendary Resistance at higher levels. That second part is a problem, especially since later features let you use this up to 5+ times.
Efreeti: Fairly meaningless at low levels, moderately good (but a very resisted damage type) at higher levels. Probably the most balanced of the 4 possiblities.
Marid: Pretty good at low levels - by level 4 it's approximately equivalent to advantage. Unreasonably good at higher levels, as it just lets you auto-succeed on a skill check.
Dao: Mediocre at low levels - by level 5 it's finally equivalent to the 1st level spell Shield. Completely broken at high levels, as a reliable +20 or even just +10 bonus to AC for a full round makes a creature functionally invincible (outside of crits, of course).

The level scaling here is a big problem. Bounded accuracy is a major design tenet of 5th Edition, and adding a reliable +20 or even just +10 to or against die rolls should not be done outside of specific situations - existing cases all shore up major weaknesses of classes, rather than just making good stuff better.
Monks get Deflect Missiles, but cannot meaningfully engage with ranged opponents.
War Clerics get Guided Strike, but are frontliners who never receive Extra Attack.
Storm Sorcerers get Storm's Fury but have to be struck by melee attacks to use it.

Miraculous Protection
Djinn: Thematically appropriate and not unreasonably powerful.
Efreeti: Thematically appropriate and not unreasonably powerful.
Marid: Thematically appropriate and not unreasonably powerful.
Dao: No, No, NO. Absolutely not. No-resource always-on resistance to the most common damage types (bludgeoning, slashing, and piercing) in the game? This would make your subclass broken if it were the only feature. And certainly not as a level 6 feature. Maybe it could be balanced in the Level 14 slot.

Mystical Mobility
Djinn: this is about the right level to get a slow fly speed.
Efreeti: A 10ft walking speed increase is actually very weak for a 10th level feature.
Marid: A swim speed is unremarkable, fairly weak for a 10th level feature.
Dao: Burrow speed is... crazy powerful compared to these other options. On-demand total cover is really good - even more so when combined with the warlock invocation Ghostly Gaze, letting you keep an eye on things even while burrowing.

Incredible Power - Strong mainly because of how good the Lesser Wishes are. Immunity to a damage type is very strong, but as long as you don't let them have it to bludgeoning, slashing, or piercing, it's appropriate for a 14th level feature. But this feature also reads as a bit scattered - most warlock 14 features are pretty self-contained, rather than massively improving previous features. Not overpowered, just kinda weird.