I'll start saying that yes, it's balanced overall, but might be weaker in combat than other races.
The only combat-relevant feature I see is the resilience against diseases.
Let us compare with other races, focusing in combat features. I'll ignore the +2/+1 ASI, as this is default for everyone. This means I'm considering your race having a +0.5 score in combat. I know speed and darkvision have some influence in combat, as well as keen senses in terms of not being surprised or surprising the enemies, but I mean from a DPR/Tankyness combat sense, or things that directly influence saving throws, attack rolls and damage.
Gnomes
Advantage against Wis, Cha and Int magic is certainly stronger than resilience against diseases. I don't think I need to argue much about it - the guide itself gives a 2 score for the Gnome.
Half-Elf
Half-elves get +1 ASI, which is a +1 score for combat. They also get Fey Ancestry, which is arguably as useful as your resilience in combat, meaning their +1 ASI is the major difference.
Half-Orcs
Relentless Endurance and Extra Crit give them +1.5 score in combat, against your +0.5.
Tiefling
Fire Resistance and Infernal Legacy again give a +2 in combat.
Dragonborn
They have the worse score in the guide, but their damage resistance is certainly more meaningful than yours, in average campaign scenarios, and Breath Weapon has its situational uses.
Halfling
Lucky by itself should be as useful as the resilience. Lightfoot has Naturally Stealth, which pretty much nullifies the -1 from size (which I already think is overestimating how bad it is to not be able to use heavy weapons or -5ft) and Stout has Poison Resilience. Both also have advantage against Frightened.
Elves and Dwarves
Both already have greater scores than other races (7+), but again, both have more combat-focused features (either +1 ASI, +1 HP/level, cantrips...)
Human
Let's be fair, normal human is a bad race. Even if the guide gives a 6 total score because it gives +6 ASI, hardly any class is actually using these to their full potential. Your race is better than normal humans, but I'd say any race is.
Variant Human
On the other hand, Variant Human might be the strongest race in the game, depending on the setting (mainly how much not having Darkvision will hurt them), since we have some strong feats. It's hard to compare since it depends on the feat chosen, but I'll argue that Sharpshooter beats +1 ASI and Resilience against diseases combat-wise.
All of this is just to say: compared to other races, you might be lacking a combat-focused feature.
Notes on the Ability Scores given and classes likely to use this race
Similar to how Normal Human is considered a bad race because +6 ASI doesn't mean much when there's no class or role that will use each score, let us quickly review your Ability Scores to see if they are indeed worth the +3 Score.
Heartwood
+2 Dex/+1 Wis: Ok, this is similar to Wood Elf, it's certainly useful for Ranger, maybe Druids. I do think Wood Elf is slightly better choice for both, balance-wise, than your Heartwood, but not enough to make it a "not a choice" scenario. (i.e. I think this is already balanced enough, but you could give an extra .5 feature to make it a more fair choice)
Woodland
+2 Dex/+1 Con: Certainly useful as well. Con is pretty much a secondary stat for every class. Dex Fighters or Rogues can make use of this stat distribution. I would say that, similar to Halflings, this would be taken mostly by Rogues.
I think the comparison between Stout Halfling and Woodland Mice depends on how strong will Darkvision actually be, compared to Lucky.
City
+2 Dex/+1 Cha: Again, similar to (Lightfoot) Halfling, so we would expect it being chosen by Rogues. Bards and Sorcerers can also benefit from it.
I think that, similar to Wood Elf x Heartwood, Lightfoot Halfling is stronger, mainly due to Naturally Stealth and Lucky, but not enough to completely dismiss your mice as a choice. Again, Darkvision being more relevant than usual can make your race get closer as well.
Conclusions
Your race is not "too weak", but I think it is slightly weaker, mainly for combat purposes, than the existing ones. You could certainly give an extra .5 score feature without making it overpowered. As a suggestion, some kind of Hiding feature, similar to Naturally Stealth and Mask of the Wild, preferably weaker than Naturally Stealth though.
As a side note, I really like the flavor in every feature you put there, and I would certainly pick it as a race for a PC of mine , even if it's suboptimal.
Animal Handling skill probably won't help
The Animal Handling skill means you are proficient in controlling and nurturing domesticated animals. There are difference between tameable, captive-bred and domesticated animals. Even when animals are domesticated, you still need skill to work with them, but that skill doesn't give you supernatural powers neither allows you to control and command any wild and/or untrained animal you can encounter.
Animal Handling proficiency can't be a substitute for any magic spell (like Dominate Beast) or class feature (like Ranger's animal companion).
If you are a player, then ask your GM
It is the GM's job to decide, what check you should make. They might ask you "make the Wisdom (Animal Handling) check" in response to you announcing "I am trying to make this bear do X" (to run away, for instance). They also might ask you for a Nature check. Or Intimidate check. Or give you Advantage because you are proficient in Animal Handling. Or ask "How do you do that?" Or simply state "the bear didn't run away".
In the end of the day, it is completely up to the GM — despite the fact wild animals usually don't cooperate with people in real world, things might differ in the setting you are playing in. If you are the GM, and you choose the check can be made, the DC should depend on the announced action. "I try to scare this bear and make it flee" should be much easier than "I try to convince the bear to go with us and fight our enemies".
Normally, players don't "use" skills
In games like Pathfinder there are various tables, describing what you can and can not do with the respective skill. In D&D 5e, however, skill descriptions are loose by design, and the reason is it's the GM who decides what check a player should make.
In 5e players aren't supposed to announce "I use my Animal Handling skill", "I use my Survival skill" or "I use my Persuasion skill". Instead, a player should describe what their character does - "I try to calm down my horse", "I search for any tracks and footprints", "I say the elf guard we mean no harm" - and the GM might ask for a ability/skill check in order to resolve the situation:
The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure.
Sometimes, the GM might ask for an ability check using a specific skill — for example, “Make a Wisdom (Perception) check.”
For example, if a character attempts to climb up a dangerous cliff, the GM might ask for a Strength (Athletics) check.
See more information in Player's Handbook, page 174 - "Ability Checks".
Best Answer
This is a place for a Charisma (Animal Handling) check
Influencing a beast to be friendly is a charisma check...
"I really want to use my animal handling proficiency though." The Player's Handbook talks about a variant rule enabling Skills with Different Abilities.
If your GM allows it perhaps you can make a Charisma (Animal Handling) check. It is certainly reasonable to include your proficiency for animal handling when interacting with an animal. Make sure to check with your GM if you are using this variant rule though.
If not...
You should use Persuasion regardless of whether the spell is active or not. Here are what the skills do according to the Basic Rules.
Animal Handling
Persuasion
"Befriending of a beast" is not within the basic purview of Animal Handling so you should use Persuasion (it would likely be close to impossible to even attempt this without Speak with Animals or a similar spell; you should at least have disadvantage since you don't share a language without the spell).
Wisdom checks overall are...
This has nothing to do with communication, which is why you cannot make a Wisdom (Animal Handling) check for this.
If your GM is allowing animal handling to be used for befriending beasts, then there is no reason to force a player to change skills with the new communication method. The intelligence of the creature didn't change, so the method for befriending would be the same; you would just get the added benefit of verbal communication (possibly advantage depending on the situation)