The verb is still in second position.
Sogar is a particle which modifies the subsequent part of the sentence. So it should be regarded as
[Sogar meine Tante] [hatte] [einen Ausdruck].
This is the only position that sogar can be in, if it should modify the subject. You can check this, by rearranging the sentence (although I admit, this is no help for a non ...
English is an SVO language which means, that in a proper English statement Subject, Verb and Object(s) appear in exactly this order. (Btw: Among all Germanic languages, English is the only one with an SVO order.)
German is not an SVO language. It's a V2 language, like all other germanic languages except English, and this means, that the Verb stands at ...
In the sentence above, auf das Wohlbefinden does not describe the place. It describes on what pets have effect. There is actually direction involved (there is some effect that goes from the pets to the well-being, so to speak). Taking this into account, accusative is the logical case here.
In my opinion it is not absolutely false, but it sounds "bumpy".
The phrase "Meine Mutter sah betrübt und besorgt aus, und sogar hatte meine Tante einen Ausdruck" consists of two main clauses (Hauptsätze). The first part "Meine Mutter sah betrübt und besorgt aus" is correct, but the problem is the second part
(und) sogar hatte ...
With the definite article, you refer to a specific device. Without an article, you refer to you might say the general concept of a device. It's similar in English:
Das Gerät wird mit dem Equipment geliefert.
The device will be delivered with the equipment.
Das Gerät wird mit Equipment geliefert.
The device will be delivered with equipment.
This is not that different from English, the problem is hard to solve. A complication, for which I could not give you a rule, is that lösende is actually an infinitive in disguise:
Das Problem ist schwer zu lösen.
Es ist ein schwer zu lösendes Problem.
Omitting the zu would, by accident, lead a grammatical sentence, but it would be non-sensical.
Es ist ...
I think the question relies on a confusion. Namely, altough worauf aludes to wo, it actually has nothing to do with it. Worauf is rather one of the wo+preposition questions like woran, worüber, wonach,... They are not necessarily related to "place", rather wo-
is a placeholder that allows you to primarily ask for the piece of information the ...