Disclaimer: Before anyone decries me for being a mean meanie who stomps on people's fun, I'd like to clarify that the OP messaged me asking me to comment, with the subject line "Please roast my skill gems :)".

I'm starting with this one because I think it has more wrong with it, and is thus more interesting to examine.

Level 20

First problem - this skill has more non-boolean stats than can fit in a single granted effect - and thus skill - at 10. The maximum is 8. Only 3 of these are coming from the base brand behaviour. There are ways to hack around this limit in some cases, but they aren't great and may cause problems down the line. Inconveniently, none of those ones are the ones that flat-out don't work (there are some, but they're boolean and thus effectively unlimited), so there's no obvious "drop these ones" here, but it's a good indication that a skill is overcomplicated.

Here's my 2nd contribution to the GGG Brand Recognition initiative.

Solid pun. I don't really have anything to say here, except that I still wish I'd been sucessful in getting Brand Recall named "Rebranding" instead - both work as a slight pun, but I really prefer the latter.

It's another Dexterity based Brand skill (like the other I designed) so the two of these new skills aren't all lumped in with Intelligence.

Boooo!

Why? Brands are pretty well set up to be primary-int, secondary-strength as a concept, themed lightly towards holiness, notably associated with Templar in general and Heirophant in particular, and very specifically - and intentionally - set up as an alternate to spell totems for that int/str spell overlap by the Runebinder keystone. Plus dex spellcasters get easier access to all the cool projectile modifiers that help them get stuff that chains between enemies in different but analogous ways to Brands.

This is pushing against part of what brands are, conceptually, and why they exist in the game - one dex brand seems fine, but we shouldn't be putting half of all Brands into their off-brand attribute (pun intended). Of the two, I think this one more justifies it's dex. I would say the other one should probably be str primary with int secondary if we want variation in brand attributes (which we might not - the socket colours make it harder for new players to swap which one they're using easily to test them out).

We primarily accomplish this by giving this skill damage over time scaling for both chaos and cold damage! Your chaos modifiers will scale any poison damage you deal

That's unrelated to the skill, though - did you mean "cold modifiers" here, based on it letting you poison with cold damage? Then it's incorrect (not all poison the skill deals will necessarily come from cold damage), but the point is at least there.

and your poison modifiers (increased poison duration, poison deals double damage, faster damage with poison, etc) will also improve its poison-like cold damage over time debuff.

Spoiler: they won't. But even if they did, less specific stuff like modifiers to chaos damage over time, or to the chaos damage over time multipler, or to ailment damage, still would not.

This is probably the correct time to address the elephant in the room:

Modifiers to Spell and Poison Damage apply to this Skill's Damage Over Time efect

There are a lot of things wrong here. First, I'm pretty sure you meant "or" there - there aren't any modifiers currently in the game that are modifiers to Spell and Poison Damage for this to affect. You also missed "also" (those non-existant modifiers should still do their usual thing as well), and missed "Damage" from Spell - there are no modifiers to just "Spell". Let's iterate:

Modifiers to Spell Damage or Poison Damage also apply to this Skill's Damage Over Time effect

Better! but very long, and it's really doing two different things in one stat. It's probably simpler to do those two things separately.

Modifiers to Spell Damage also apply to this Skill's Damage Over Time effect

Modifiers to Poison Damage also apply to this Skill's Damage Over Time effect

The first stat there is fine - we've done that, that works. The second one runs into another issue, which is that there are no "Modifiers to Poison Damage" in PoE. There are, however, modifiers to "Damage with Poison". And unfortunately, just writing:

Modifiers to Damage with Poison also apply to this Skill's Damage Over Time effect

Isn't actually going to fix anything. Those modifiers aren't differentiating on anything about the damage, they're about how it's dealt, or more specifically, that it's part of the poison damage calculation - coming from the base damage of the hit and going through the poison damage calculation. That's not something we can just change - the poison damage calculations alone are 5660 lines of code, and they're set up on the fundamental assumption that what you're trying can't happen. Modifiers to Damage with Poison really can't apply to anything that isn't Poison without a massive refactor of those systems. Also, you're including things that can apply to Poison by nature of it being an ailment and thus part of hit damage calculation, which is very different from regular damage-over-time calculation.

"Spell Damage" is one of the four damage methods, and it's modifiers are a well-defined set that are gathered together, and can't possibly have any overlap with modifiers to a skill's damage over time effect. None of that is true of modifiers to "Damage with Poison".

The overhead in work here, as well as the ongoing issues with maintenence of the extra weird calculations we'd have to do, far outweigh the potential benefit even without factoring in that it will still be unintuitive to players in a bunch of cases even if we do it.

I gave it a radius of 6 which is half the size of Arctic Breath's radius of 12.

Arctic Breath's radius is 15.

Any modifiers to poison will affect the stacking cold debuff / damage over time effect:

That's not what the gem says. Even that was limited to modifiers to damage with poison. You're definitely not getting any of this other stuff applying (or rather, some of it could and some of it couldn't, but you'd actually need a stat which made it apply for each kind of modifier).

Noxious Strike

50% increased Damage with Poison

Can't be made to apply for reasons noted above. The other stats aren't modifying damage with poison in the first place, but will affect the poisons the skill causes.

+25% to Damage over Time Multiplier for Poison

For the record, this is not a modifier to Damage with Poison. The Damage over Time multipler is a modifier to damage with poison, and this is a modifier to that, not to the damage with poison iteslf. Similar to if there was a modifier to the effect of poison.

Toxic Delivery

Poison you inflict with Critical Strikes deals 50% more Damage

This is one of those things I was mentioning that Poison can do due to being an ailment. There's no way for a stat to apply this kind of modifier to a non-ailment DoT, because unlike the hit/ailment calculation, those are pure stats and don't support conditional modifiers of this type. The only way to accomplish this would be for the skill code to manually fake it, and then we've got a stat that someone will try to use elsewhere months or years down the track, and it won't actually do the things it says it does, because it's relying on a hack specific to the skill. If the issues with applying Damage with Poison modifiers to skill dot we to magically disappear, this would still be a potential issue - we could do it, but it wouldn't be great to do so.

Long term, support for such conditional modifiers on regular DoT is something we'd like, but it's quite challenging for a variety of reasons.

+0.4% to Critical Strike Chance per Poison affecting Enemy, up to +2.0%

Again, clearly not trying to apply.

Void Beacon

None of these apply directly to anything, but the cold resistance penalty helps the cold dot, and the chaos res penalty helps the poison. Sure.

Withering Presence

+20% to Chaos Damage over Time Multiplier

As above, this isn't a modifier to damage, so even if we could do your stat, it wouldn't make this affect the cold dot. It would affect the poison. Other stats on this one are not applicable.

Frigid Wake

+20% to Cold Damage over Time Multiplier

This one of course affects the cold dot, none of the others apply.

So a lot of the interactions you want for the cold dot aren't going to work, and as noted at the start, the skill has more stats than we can actually stuff into it. As such, my recomendation would be to drop the base tertiary duration (which you've got labelled as just "base duration", which is technically incorrect - base duration and base secondary duration are the attached/detached duration on brands), and the base cold damage per second, and as such, drop the separate cold debuff entirely - because it really can't do the things you want it to do. This would mean also dropping the stat that was trying to make spell/poison damage modifiers apply to said dot, but that's boolean and doesn't free up any space.

Poisoning with cold damage already gives the skill a DoT component that's interesting and mostly unique, having a second, very different one on top of that when it'll be hard to scale both (both because your desired mushing-together of fundamentally different modifiers won't work, and because it misses a bunch of stuff). This conveniently removes the two stats we need in order to actually make this a skill, and maybe later you can suggest a support gem that causes skills to add a cold-damage debuff to whatever they poison, in a manner similar to Decay Support.