The formatting on your post is really hard to follow, so I put this together so that it was slightly more legible. I also adjusted as much of the wording as I could without interfering with the effects to be in line with standard magical items in the 5e system.
Now that I can parse your items, I'm going to give you a few notes on them.
This staff has charges but does not have any method for regaining them, meaning it is consumable (and thus much less powerful).
When listing spells that a magical item can cast, they should be in alphabetical order, not in a loose order based on the number of charges they take to cast.
Make sure to specify what statistics they use; if you don't, they cannot be cast confidently and you'll have to make a ruling every time someone tries to use the item (I have fixed this in the link above).
When making a magical item that can cast spells, resist the urge to go through the spell list and just pick any spells you like and throw them on there. Your item should have a cohesive theme and stick to it; a staff of fire can only cast fire-related spells, a staff of healing can only cast healing and restorative spells, etc.
For immensely powerful spellcasting staves (like this one), try to keep the spells chosen from one class' spell list to maintain cohesion. The Staff of Seasons has spells from the wizard, cleric, druid, and warlock spell lists, any many of them do not apply to the "seasonal" theme that you are clearly trying to go for. Haste, armor of Agathis, and feather fall do not belong, to name a few.
For Warmth of the Heart, you cannot assume that people know what you're talking about, that they know what you mean, or any other assumptions. Specify what you mean by "fire" and "life" spells. If fire spells deal an extra die of damage, say "When you cast a spell that deals fire damage, increase the number of dice you roll for damage by 1." If healing spells restore an extra die of hit points, say "When you cast a spell that restores hit points, increase the number of dice you roll for healing by 1." Be specific. The more clear that your item is, the less overhead you have during play.
For One with Nature, it's always risky to include an entire class feature in a magical item, especially one with fewer restrictions than the base version. When possible, use the phrase "as with the druid's Wild Shape feature" or "as with the spell polymorph"; doing so directs any rule inquiries towards those features and spells, rather than requiring the item to detail every restriction.
Balance assessment: Definitely a legendary quality item. The spells that it can cast are too varied, dipping into too many spell lists and allowing too much freedom between healing and damage dealing. However, if it was intended that the staff has a limited number of charges (I doubt it), then the weapon's power comes down significantly. I would still classify it as legendary, but it's no artifact.
Overall assessment: This item seems, to me, like something that was made after having a neat idea about combining several items into one super powerful one and assuming that "legendary" and "artifact" can be put together to mean "no limitations." This item should never be given to a player.
Once more, charges but no method of recharging. Legendary items (let alone artifacts) are only very rarely consumable, do so with caution.
For a legendary item or an artifact, a +1 bonus to physical attacks on a staff is pitiful.
Bridge of branches seems to be a custom spell, and because of that I cannot say for sure anything regarding the balance of this item. If it simply creates a bridge of 60~ feet or less made out of branches for 1-10 minutes, it's a reasonable 2nd-level spell.
Balance assessment: It's not nearly a legendary item. If the charges are intended to recharge, and do so in a volume that does not fully recharge the item every day, I would say this is a very rare item at most.
This is, at most, a rare item. I would lean towards uncommon if it was Wisdom (Nature) checks regarding only plant life.
My assessment assumes that the "bloom" effect of the item's property is essentially a druidcraft spell effect. This uncertainty means that it needs to be more specific.
Balance assessment: This item has almost no effect on balance.
Overall assessment: Granting a player advantage on all checks of a given skill can be potent if the campaign makes use of that skill a lot. But is it even fun to use, or does it remove some of the entertainment in not always succeeding? At what point do you just stop making them roll to determine something and just assume they know it? This item seems a bit shortsighted to me; the goal isn't to just make them succeed more, it is to improve their options. This item is a passive benefit at best, and so very boring.
This is a frost brand weapon straight out of the DMG, with one effect removed and another added. As the base weapon is a very rare weapon and the changed effects are not overwhelmingly powerful, this is still a very rare weapon - not nearly legendary, and certainly no artifact.
The Chill of Winter effect is a neat idea, but I don't know if "any liquid" is really appropriate.
Balance assessment: No better or worse than a standard DMG item.
It's an item that improves your spellcasting and your unarmed attacks? These are two very different things, and only an incredibly small subsection of characters will make use of both. Most spellcasters would actually be handicapping themselves by using an unarmed attack, but this might be good in a spellcasting monk's hands.
As mentioned above, the Warmth of the Heart property needs to be more specific. On its own, it's a pretty boring idea, but it's certainly effective.
You need to be specific about the benefits it grants. You gain advantage against that creature? On what, saving throws? Attack rolls? Everything? For how long does that last? I made my best guesses when I formatted it, but they could be very wrong since the original item was so vague.
Balance assessment: Depending on the actual effects of Warmth of the Heart, this item is at least very rare. I hesitate to call it legendary, but in terms of raw (uninteresting) power, it's around there...ish. In a very specific circumstance.
As mentioned above, granting less-restrictive class features with an item is really iffy territory.
At first glance, this double-dips AC bonuses, which is really bad. But in actuality, you don't get the benefits of your armor while in another form, so having this specify that your beast shapes have improved AC is a good idea.
Balance assessment: Legendary, but only on account of granting a full, less-restricted class feature on top of +2 AC at all times.
Overall assessment: You want to avoid a situation where a single item is better at being a specific class than a high-level character that has all of their levels in that class. Consider restricting attunement to druids only and reducing One with Nature effect to only slightly improve Wild Shape's natural restrictions. Just by increasing the maximum challenge rating of an attuned druid's Wild Shape by 1 can vastly improve the effect of the feature. As it stands, One with Nature allows any character (with any combination of class levels) to wild shape into a T-rex at level 16. By comparison, the most potent wild shape a druid can have (20 druid levels, circle of the moon) is a full 2 CR below that.
(con't)