a) When the Druid loses her Wildshape form, the Spirit Bond/Spirit animal disappears. Looks like the game thinks the Druid died.
The Spirit is gone until I take a Long Rest, so really breaks the Wildshape option for a Druid of the Kindred Spirit.
I am not the first to report this, but others were not reported on the forum if my memory is correct.
b) My Spirit Wolf is doing only 1d6 damage without the Wisdom bonus, though not all the times. I am pretty sure it is when my Druid is transformed in a bear for example. 
Druids maintain their mental stats while in animal form in 5e. This issue is frustrating, as my Wolf is my permanent companion till the end of the game, so 1d6 damage is pathetic, for the real risk of the Druid taking 3d6 damage and being stunned when the Spirit Wolf falls.
UPDATE : I noticed, out of combat and in Humanoid form, that my Spirit Wolf has +5 to hit, but +0 to damage. My Druid has +3 to Wisdom at the time.
c) EDIT: I removed this point, as I finally found how to see the Spirit animal HP. Instead of hovering the portrait info, you can click the animal as such (out of combat anyhow). The HP total was fine and was not affected by Wildshape.
Fix suggested : removed my bit here. Point A and B are bugs.
Steam profile : https://steamcommunity.com/id/baraz/
Addendum :
- Version : 1.2.11 Public Beta [identical to 1.2.11 hotfix as Steam did not update]
(My original DLC was beta, which I am told should not be a factor.) - no mods.
Steam profile : https://steamcommunity.com/id/baraz/
Confirmed (after testing) : when summoned, the Spirit Wolf has its bonus damage (Wisdom bonus from Druid), but loading a saved game reverts the damage to only 1d6.
On Discord, other players reported the Spirit Wolf not doing the bonus Wisdom damage.
- Which is an issue as it is the main feature of the Circle of druids.
One person noted "It seems to do the extra damage when you resummon it, and lose it after loading".
Based on another comment, it seems loading a save does not load the proper the Spirit damage.
Nb : my tactic now is to let my Spirit animal die before reaching a new area or before ending a session. :P
Steam profile : https://steamcommunity.com/id/baraz/
I think I should not be able to summon my spirit again after it died on the same day.
However I can do so, when I'm in Wild Shape form. The spirit doesn't use the proper modifiers then, but is obviously still useful for tanking etc.
If I could dismiss it and resummon it, I could bypass the current bug. :P
Steam profile : https://steamcommunity.com/id/baraz/
New update with Public Beta 1.2.12 Hotfix :
To my surprise today, my Spirit Wolf has its bonus damage using both a hard save or quicksave in Caer Cyflen.
The bug was ongoing yesterday using the same version of the game, so I am a bit puzzled ...
Will report again if I figure how to reproduce 100% the bug.
[ EDIT: after traveling the Spirit lost its damage bonus again (it was OK after I exited game and loaded a hard save in Caer Cyflen).]
Steam profile : https://steamcommunity.com/id/baraz/
For the Spirit animal bug (damage bonus vanishes after traveling and loading a save), I found a way of re-summoning it:
I cast Flaming Sphere on it (or other AOE). Not the most immersive method, but you then re-summon it with its correct damage.
The difference between 1d6 (average 3.5) and 1d6+5 (average 8.5) is over 100% on average.
Steam profile : https://steamcommunity.com/id/baraz/
Patch (beta) 1.2.14 mentions fixing the Spirit animal damage (from Druid's Wisdom) while in Wild Shape (Fixed Kindred Spirit not correctly using the Druid's stats when summoned while in Wild Shape).
While tested as working as such right now (I only looked at the damage: did not check the HP which I presumed was fine), when I reverted to humanoid form, my Spirit Wolf lost the damage bonus from the Druid's Wisdom. Though the cause of that bug may be a different thing altogether. It is true that, in Wild Shape, my Spirit Wolf was summoned with the correct stats from what I saw.
nb : sorry for posting so much today.
Steam profile : https://steamcommunity.com/id/baraz/
Also spirit animals should be immune to poison which they are not and they receive the damage from poisonous sources.
I have encountered this bug on the most recent patch (Version 1.3.55) with completely a clean Mod-less game. My Wolf Kindred Spirit loses it's (Druid Wis) bonus to damage regularly in the course of normal gameplay.
Seems very easy to reproduce, from what I can tell it seems to either:
1) Happen on any save file LOAD
2) Long resting
3) Reverting from Wild Shape
Current workaround seems to be unfortunately killing off my Kindred Spirit and re-summoning it, which will give it it's (Druid Wis) bonus to damage back.