[RPG] Can a Warlock become a lich

The Warlock's spell slot level indicates the maximum spell level they may select.

I agree the wording is a little confusing but consider the general rule for spell slots:

When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell. You can think of a spell slot as a groove of a certain size--small for a 1st-level slot, larger for a spell of higher level. A 1st-level spell fits into a slot of any size [...]

In other words, the phrase "for which you have spell slots" more fully means "for which you have spell slots that the spell can fill."

The rules for Warlock spells known suggests newly gained spells for a Warlock may choose spells that are no higher than what is shown on the table for their level:

The Spells Known column of the Warlock table shows when you learn more warlock spells of your choice of 1st level and higher. A spell you choose must be of a level no higher than what's shown in the table's Slot Level column for your level.

Additionally, when you gain a level in this class, you can choose one of the warlock spells you know and replace it with another spell from the warlock spell list, which also must be of a level for which you have spell slots.

The wording here explicitly states that newly learned spells must be "no higher" than the slot level indicated for your current level on the warlock table.

The subsequent paragraph goes on to say that if you swap a spell out for a new one, that new spell must "also" be of a level for which you have spell slots. The word "also" indicates that it follows the same rules for the new spells (ie, equal to or less than the slot level on the table).

Another way of thinking about this: Classes that must prepare spells from a larger list (Druid, Cleric, Wizard) are told "The spells must be of a level for which you have spell slots." It's the same "for" wording, but no one would argue that these classes must prepare spells of equal (base) level available to them. They are free to choose only lower level spell and upcast them using the available higher level slots.

Altogether, the rules indicate that a Warlock is free to choose any spell that is of a level that is equal to or less than the slot level indicated on the Warlock table, regardless of whether that spell is swapped in or learned in addition to the previously known spells.

This sub class needs some work before it is balanced with the others

There are a number of issues with this, but I'd say it's simply too strong, and I think some of the other issues needs to be addressed. I compare to "Dark One’s Blessing" because a lot of the features are of similar mechanics.

Impelling Shroud

This is strong.

1. Compare it to Dark One’s Blessing, which gives you level + Cha temporary hit points. They seem comparable, but there's a major difference. The temporary hit points are lost after a long rest, while these are regained. You need to have multiple encounters per long rest before the other one is better.

2. Next thing is that with impelling shroud you start with the hit points, where as with Dark One’s Blessing you have to kill something first, so that one needs multiple enemies to be as effective.

3. With Dark One’s Blessing you need to be the one dealing the killing blow to even get it. All of this combined makes impelling shroud much stronger.

Sacrificial Virtue

This one is what I would consider the worst designed feature and I suspect that you care too much for the positives to realize all of the negatives.

1. The other level 1 feature was strong, so doubling down and giving an additional one is too much. If you truly want this, you need to remove something else and give this instead.
2. This is blatantly overpowered. Instead of nothing, you now have the ability to give half/level D6 multiplied with players damage. This is MUCH stronger than nothing.

3. Lastly, and this is important. The design is somewhat broken. You have added a price (Sacrifizing a creature that's goodish). The issue with the price is that for some characters that wouldn't even be a price at all. Now I know you say it's a big price for your players, but that doesn't make it much better. This is like saying: "Either you go against your characters wishes or you won't get to use this at all".

   So either it's directly useless, which feels bad as a player or it's insanely overpowered, which feels bad for everyone.

I know it's meant as a "last resort" to your player, but you need to find a price that's equal to most characters and feels like an extreme price to pay, but not one you will feel bad about and not one that does specifically against the personality of the character/party.

Dark One's Own Luck

Not much to say. This one is fine, but I would suggest removing it to make room for the reworked version of Sacrificial Virtue.

Embalming Presence

I like your fluff, but I have no idea how this makes me feel like an evil character sowing fear among my allies. I'd feel like a good guy buffing most of my party.

1. This is again too strong. Compared to Fiendish Resilience this is much stronger. How often will Fiendish Resilience save you enough health that it matches your level + cha modifier. I'm willing to bet not too often.
2. This gives that much to your allies as well. So we really need to ask, how often does Fiendish Resilience save enough health that it matches four times that much. I'm willing to bet that number is very close to never.
3. You already gave them an ability that gives them health, don't give them even more. I'd suggest reworking this ability into something that gives a cool effect, but each long rest, a random character within 100 feet takes XD6 of damage. This would create the evil guy feat effect where your party pays the price for your power. It's not perfect, but it matches the theme better.

Deaths Inevitable Embrace

This is poorly designed. I like the execute effect, but it has insane synergy with Sacrificial Virtue. With that many D6 worth of damage and this ability, there's a fair chance you'd one-shot even some level appropriate enemies that are meant to be a challenge for the entire party.

The design problem comes from your own words: "Potentially quite powerful but very specific in its use", this is bad. If something is powerful and fun, I want to use it! Abilities like these will go one of two ways:

1. in some combats they will be useless, which will feel bad, because the player probably wants them to happen.

2. In some combats they will dominate, which won't feel good either.

   Compare it to Hurl Through Hell. Hurl Through Hell is obviously stronger when it happens, but it's also "specific in its use", that specific being once a day. That is much better. That way the player will get to use it regularly and it won't suddenly pop multiple times during one encounter.

I'd suggest turning it into some kind of once-per-long-rest ability, as this gives the player much more agency. Maybe something that can only be activated when someone else thinks you are friendly to them (As means of assassination)

Final Thoughts

I would suggest moving some things around, nerfing the class generally and making it focus a little more on utility. This could be utility that helps isolate and kill targets.

This class takes too much agency away from the players and gives too much power. I think there are some really cool ideas, so I think you should keep working on it! This is definitely a cool start.