Overall a good start
First impressions of these spells as a whole is that they are a good start and fit the theme you are going for nicely. That said I think there might be a few issues worth mentioning, so lets go through each spell in detail.
This is basically a situationally triggered version of misty step, which is a second level spell. But yours is a reaction instead of a bonus action and comes with a damage boost. It is also similar to the Way of Shadow monk's sixth level feature Shadow Step in how it is a limited teleport.
In power terms I think this spell is alright, if leaning slightly to the more powerful side for a first level spell. But given it's limitations it would likely be underpowered at second level.
One potential useful exploit would be if someone capable of casting revivify were to take this spell, instant teleport to the side of a fallen ally could be quite useful but not game breaking.
Drain the Dying
This spell is a bit overpowered, particularly when upcasting it. The damage portion is fine since damage effects from the same source can't stack and triggering a bit of extra damage everytime someone dies nearby is quite cool. The broken portion is the non-concentration, unlimited continuous healing.
With this spell cast you could run into a hoard of enemies and cast fireball centred on yourself, provided you kill at least a couple of them your would gain hitpoints from doing so. This is clearly an overpowered combination.
The closest spell I could find to this is Vampiric Touch a third level spell that only lasts 1 minute and does require concentration. I would consider rewriting this spell with that as a starting point, at least with the reduced duration.
This is a nice spell but needs a little work. I think is would be reasonably balanced with a few tweaks.
First, I think you need to give a save against the effect of the spell. A melee spell attack is a lot easier land than it is to make a powerful enemy fail a save. By having an attack roll and no save until the end of their turn you make this almost an auto-succeed effect that is extremely powerful.
By preventing spells until the end of their turn, they can't counterspell anything your party does, or use what are presumably their best abilities for a whole round. A single casting of this effectively gives your party 2 rounds to wail on the enemy with little chance of them hitting back.
Second, simplify the mechanics. Currently this spell triggers multiple rolls per turn; an attack roll, a concentration check for the spell effect, concentration check from the damage taken and then save against the spell at the end of their turn. A single save per turn and more clearly worded effects would be an improvement.
Third, ditch the damage part. This spell is already pretty strong and the damage isn't really significant anyway.
Finally, add a duration. You have missed giving this a duration but I would suggest 1 minute would be balanced, though the spell is unlikely to ever last the whole duration.
With these changes the spell would look something like:
Level 3: Abjuration, 1 action
Duration: 1 minute
You send out a wave of psychic dissonance toward a creature you can see within range. The target must make an intelligence saving throw, on a failed save the targets concentration (if concentrating) is broken and they are unable cast spells for the duration. They may repeat this save at the end of each of their turns, on a success the spell ends.
This version still isn't perfect and probably needs more balance tweaking to get it right but it is much simpler while still achieving the same effects.
A final word
One note that could apply to any of these spells and your whole theme in general. Spells and abilities that benefit from death, particularly ones like drain the dying can quickly turn the tide of battles and result in death spirals. When used by a PC this isn't terrible and can sometimes even be desirable, but if a DM plans to use spells like this against the players they should be wary of TPKs.
Watch out for these spells trivialising some combats during your playtesting and make adjustments where necessary. Even individually balanced spells can get out of hand if a player can combine them cleverly.