Note: I'm not experienced at balancing homebrew content. I am simply comparing these items to official (DMG) material assuming it is balanced.

Item 1 is overpowered

Anti-magic field is an 8th level spell. an 8th level one-time use scroll according to DMG page 200 is a very rare item. So one that refreshes every day should be at least legendary, but even that might be too powerful.

Item 2 is inconclusive

there are several wands, such as wand of fireball that have up to 7 charges of 3rd level spells and recharge 1d7 of them at dawn, so that would suggest that 3rd level counterspell that has only 4 charges and recharges 1d4 of them would be underpowered. But that does not take into account action economy - to use wand of fireball you have to use an action, but counterspell can use reaction. Given the fact that most of the time most characters don't get an option to use reaction that makes this item much more powerful. I haven't found a good analogue for this in official materials, so the value of 3rd level spell charges vs 'free' reactions is debatable.

Item 3 is at least rare

The closest analog to this is Mantle of spell resistance which is rare and states:

You have advantage on saving throws against spells while you wear this cloak.

First we have to evaluate the difference in benefit on all spell saving throws vs Dexterity saving throws. If we assume that spells are equally likely to use Dexterity, Wisdom, Constitution or one of the other stat saving throws that means this artifact is about 1/4rd as good as Mantle of spell resistance.

Second we have to evaluate the relative power of the effects. If the saving throw is likely to succeed 50% of the time, this item will reduce damage by average of 75% (50% on fail and 100% on success) while the Mantle of spell protection will only reduce damage by average of 12.5% (25% of time(if first roll fails but second succeeds) by 50%). That makes the effect of this item 6 times stronger than Mantle of spell protection.

As a result this seems to be about 50% stronger, but less versatile item than Mantle of spell resistance, so it should be at least as rare if not more so.

Item 4 likely underpowered

Currently it does not specify DC of the saving throw, that makes it impossible to tell how often this ability will come into effect, but generally if you are fighting in a party or with minions and the caster has more than one enemy in range, the usefulness of this ability would be quite limited, because choosing the second best target for a spell is usually almost as good as choosing the best target. There might be some enemies that have only limited ranged spells, but even then the fact that the effect and on any aggressive action mean that the item is useful in only very niche circumstances.

Item 5 is probably ok

As you said there are no equivalents of this item anywhere in the official rules as there are no items that guarantee breaking concentration, especially not for a group of creatures. But there is a spell Magic Missiles which makes the target roll 3 concentration checks without any chance to avoid them, and you can get it from uncommon item Wand of Magic Missiles with 7 charges 1d7 of them recharging each day. Changing 3 concentration rolls to guaranteed failure, removing damage, but giving area of effect (which might not be that useful unless casters are plentiful) while increasing rarity seems quite fair.

Brooch of shielding is balanced

My method of evaluation is comparing items to official content. As Brooch of shielding is official all I can say is that it is balanced

Item 7 gets better with tougher foes

Evaluating this item is tough because while it seems very similar to armor of resistance and magic weapons seem to be about as frequent as some of the damage types the problem is that magic weapons are mostly possessed by higher level opponents unlike damage types which are distributed more randomly or based on theme. That means that this kind of armor would be next to useless for one facing low level mundane enemies such as brigands and orcs, while being extremely useful in battles against fiends and high level adventurers (unless they have backup non-magical weapons).

Item 8 seems close to balanced

Looking at Legendary items that provide protection against spells there are Ring of spell turning, Robe of Archmagi and Scarab of protection - all of which give advantage to saving throws, but also have additional effects. While resistance is clearly superior when targeted by spells that are used primarily for dealing damage, most of them can also be partially mitigated by succeeding on saving throw while resistance does nothing against crippling debuff spells. In the end resistance vs advantage on saving throws seems quite close, so this item seems quite balanced.

No, with the latest errata and clarifications from paizo available here. In fact, this specific interaction with Conduct Energy is called out in the section for the Lost Omens Ancestry Guide that the ability comes from:

What actions qualify for the requirements of the resonant weapon trait's Conduct Energy action?

You can only use Conduct Energy with actions that have one of the required energy traits. Using an action other than a spell that causes damage with the listed trait does not necessarily qualify unless the action also has the trait.

For example, if you used the ifrit's Scorching Disarm action, you could channel fire energy into your weapon via Conduct Energy, as Scorching Disarm itself has the fire trait.

However, if you made a Strike with a flaming weapon, the Strike action does not have the fire trait, so you couldn't use Conduct Energy.

So Strikes made with rune-etched weapons don't inherit the associated elemental trait. Such Strikes are still considered magical, by this line under the Damage Types section:

Furthermore, most incorporeal creatures have additional, though lower, resistance to magical physical damage (such as damage dealt from a mace with the magic trait) and most other damage types.

Interestingly, the original printing of the Core Rulebook had a contrary section to this on page 451, Damage Types and Traits, quoted below. The section was removed silently in the second printing.

Damage Types and Traits When an attack deals a type of damage, the attack action gains that trait. For example, the Strikes and attack actions you use wielding a sword when its flaming rune is active gain the fire trait, since the rune gives the weapon the ability to deal fire damage.

Finally, you wouldn't deal quite that much damage in any case. As @WeirdFrog mentioned, conducting weapons only deal an additional 1d8, not xd8 as you've suggested.