This feat is not balanced in its current state.

You're basically making a better version of the Healer feat, in every way. And also giving ability increase. There's literally no reason to take Healer with this feat available, which should be a tip-off that something isn't right.

No limit on the 1d4+proficiency (starts at 3-6 healing, ends at 7-10 healing) removes the need for healing potions (2d4+2 healing). Reminder: Healing kits are 5gp for ten uses; Healing Potions are 50gp for one use.

Most enemies are humanoids or beasts, so it's a huge boon to combat damage. (in that it applies to almost every situation)

And then, a bump to Strength or Wisdom.

Let's look at existing feats.

Blade Mastery
Unearthed Arcana 6 June 2016

You master the shortsword, longsword, scimitar, rapier, and greatsword. You gain the following benefits when using any of them:

• You gain a +1 bonus to attack rolls you make with the weapon.
• On your turn, you can use your reaction to assume a parrying stance, provided you have the weapon in hand. Doing so grants a +1 bonus to your AC until the start of your next turn or until you're not holding the weapon.
• When you make an opportunity attack with the weapon, you have advantage on the attack roll.

--

Healer
You are an able physician, allowing you to mend wounds quickly and get your allies back in the fight. You gain the following benefits:

• When you use a healer’s kit to stabilize a dying creature, that creature also regains 1 hit point.
• As an action, you can spend one use of a healer’s kit to tend to a creature and restore 1d6 + 4 hit points to it, plus additional hit points equal to the creature’s maximum number of Hit Dice. The creature can’t regain hit points from this feat again until it finishes a short or long rest.

(emphasis mine)

Blade Mastery hasn't been "okayed" yet as an official source, but we'll use it for the sake of comparison on how to value things.

+1 damage and +1 to hit are vaguely equivalent, with to-hit maybe being slightly more valuable, but we'll go with it. This is worth slightly less than half a feat. (+2 strength would give +1 damage AND to-hit, as well as saving throws, etc)

The extra, cheap, nonmagical healing option listed for healer is worth about half a feat (probably more). The only source is other feats. Magic initiate, for example, would allow you ONE use of cure wounds, compared to once per character.

And a single ASI is worth half of a feat, by virtue of the standard being two ASIs.

What should you do?

I'm not sure why you don't just take Healer and call it a day.

Barring that, I would pick two of the three options and put a limit on the healing, other than half hit points.

Perhaps:

Vivisectionist

flavor text

• You gain a +1 bonus to attack rolls you make against Humanoids and Beasts. (or damage, whichever)
• As an action, you can spend one use of a healer’s kit to tend to a creature and restore 1d6 + 4 hit points to it, plus additional hit points equal to the creature’s maximum number of Hit Dice. The creature can’t regain hit points from this feat again until it finishes a short or long rest.

Spell Mimic is too flexible

I would not say that:

This feat is basically a watered down version of the arcane tricksters 17th level Spell Thief ability.

Because in many ways the Spell Mimic feat is better than the Spell Thief feature:

• The Spell Mimic's trigger is less specific than the Spell Thief's.
• The Spell Mimic cannot fail, whereas the Spell Thief can (and does more often than not).
• The Spell Thief's ability to reproduce a spell is implicitly limited to 4th level spells slots, whereas the Spell Mimic can even work with 9th level spells.
• The Spell Mimic recovers on short rests while the Spell Thief does not.

I'll grant you that in some respects Spell Mimic in inferior to Spell Thief:

• Spell Thief preserves the memory of the new spell for much much longer than Spell Mimic.
• Spell Mimic is implicitly useless with spells that have a casting time greater than 1 minute, whereas Spell Thief is not.
• the Spell Mimic may not use their prefered spellcasting ability, whereas the Spell Thief always uses intelligence.

But Spell Mimic also lacks effects that make Spell Thief better against foes and inefficient with allies:

• Spell Thief negates the triggering spell, whereas Spell Mimic does not.
• The target of Spell Thief forgets the triggering spell whereas the target of Spell Mimic does not.

This means Spell Mimic is best suited to reliably double the efforts of your allies instead of "using an opponent's strength against them".

In practice, Spell Mimic gives a caster temporary access to the spells prepared by the rest of the party when they are most needed and effective. This flexibility combined with some teamwork makes Spell Mimic unbalanced.

Whenever the party really needs something (whether it be heals, buffs, control, or raw damage) as long as at least one other caster can provide it, the character with Spell Mimic can double their efforts by mimicking their most suitable spell. Whereas it is much rarer for an opponent to use a spell worth mimicking in any given circumstance.

Then there are also class-specific shenanigans, like:

• A Warlock can use buffs (especially ones with long duration) and heals they usually don't have access to before a short rest.
• A Sorcerer can make up for their notoriously small amount of spells known.
• A Wizard can create contingencies of mimicked spells.
• A multiclass of two casters can mimick spells of a higher level than they can learn.
• A Tempest Cleric can spam a couple max-damage chain lightning.
• Etc.

My bare minimum recommendation to balance Spell Mimic is to somehow discourage targeting allies and limit it to spells of 5th level or lower.

However, it may be easier to achieve a feat that uses an opponent's strength against them via modifying something less convoluted than Spell Thief. I would explore an effect similar to the Ring of Spell Turning with adjustments like removing the advantage on saves and limiting which spell levels can be turned to match the growth of a half-caster.