I made a designed a warlock patron called the Lady of Pain. Is this subclass balanced against the other official warlock subclasses?

The Lady of Pain

You have quietly made a Pact with the Lady of Pain and her Cage. As a
shadow of her serenity, you have been charged with maintaining balance
among the planes while preserving the Lady’s privacy. As part of this
pact, you have the following restrictions & benefits:

• You must never divulge the nature of your pact, even under pain of
  death.


• You must protect the Dabu and the city of Sigil, and never bring either
  harm.


• Alcohol and other intoxicants have no diminishing effect on the pain
  you feel. Conversely, your tolerance for pain has increased by
  twice that of most mortal creatures.


• The City of Doors is intuitive for you to navigate; most doors there
  are available for your travel if they are not being actively
  obstructed or hidden.


• You can understand the strange visual language of the Dabu, but cannot
  communicate using it.

Expanded Spell List

1st: Shield, Inflict Wounds
2nd: Cloud of Daggers, Heat Metal
3rd: Haste, Glyph of Warding
4th: Greater Invisibility, Death Ward
5th: Modify Memory, Dispel Evil And Good

Features

Serenity’s Shadow

Starting at 1st level, you can cast an ominous shadow in a 10-foot
radius that provides you with several benefits:

• You gain a bonus to your AC equal to half your Charisma modifier,
  rounded down (minimum of +1). You also have advantage on Dexterity
  (Stealth) checks.


• As part of a melee attack, you can turn the shadow against hostile
  creatures, rendering them vulnerable to slashing, bludgeoning, and
  piercing damage until the end of your turn. Once the shadow is used
  in this way, it recedes until you have taken a short rest.

At 14th level, the shadow’s aura protects and hides allies with its
radius, which also expands to 30 feet.

Severance

By 6th level, your body has adapted to recovering from gruesome
injuries. If you have hit points equal to or less than half your hit
point maximum, you can use a bonus action to regain hit points equal
to 1d4 + your Charisma modifier. In addition, when you use this ability
while holding a severed body part (other than your head) to where it
fell off, it reattaches.

Once you use this feature to reattach a limb, you can’t use it again
until you finish a short or long rest.

Planar Doorman

At 10th level, you can cast the Plane Shift spell once without using a
spell slot or requiring material components. You can also immediately
identify the plane and general location of someone is who is using a
spell or magical effect to travel to another plane. As an reaction to
the Plane Shift spell being cast by another creature, you can counter
that spell immediately.

Using the feature in either of these ways will require a long rest
before using either functionality again.

The Lady's Maze

At 14th level, you gain mastery of the specific pocket dimensions
wherein the Lady of Pain stuffs Berks who displease her. You can cast
the Maze spell once without using a spell slot. The maze appears to be
a circular series of platforms, paths, and portals.

The maze can also be cast on willing group of creatures equal to your
Charisma modifier. items and inanimate objects left in the maze will
remain there. If the maze is used this way, you must succeed a DC 20
Intelligence check to end the spell early.

Once this feature has been used for either function you must take a
long rest to gain access to the maze again.

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, 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.