I’ll try to give you more insight into your problem, as I can see it better after having read the comments.

1. As said in the comments there is no contention as to which form is used most because the first one you mention is not at all used in the language. So all that you dispose of is « Cette ne maladie se voit pas souvent chez les chiens. », which but for the slight error of an intrusive “ne” is in the fringes of the language.
   It is incorrect even without the particle “ne”, although a precision should be added; it is so in the written language (but possible in a letter to a close friend for instance). However, when you use this form in the spoken language you will not pass for a careful speaker; you can say it and there is a certain part of the French population that will say it; it might sometimes even be heard in the mouth of the not so ill educaded.

examples:

Ça se trouve pas dans toutes les boutiques, il faut chercher.
La journée de travail se divise pas en trois parties mais en deux.

2. You are confusing “pronominal form” with “passive form”; what you have in your example is the pronominal form because the verb is conjugated “à la voix active”, that is to say that the conjugated form is “trouve” whereas in the passive voice you’d have “est trouvée“. Note that at least in French, a pronominal verb cannot be conjugated in the passive voice; in the example I give we are dealing with two different verbs (« se trouver » et « trouver »); this was done for the sake of giving an example of the different auxiliary words as the passive form does not exit.

3. As told you in the comments “maladie” is the subject and there can be only one of those (which is a basic notion in all languages); of course there is a simplification involved in saying that; this is because the main word in what is called the “group of the subject” (cette maladie) is “maladie”. The agent, as you call it, or in French “le complément d’agent”, is not at all a concept associated with the pronominal form; it is only found as associated to the passive form and with some other questions of lesser importance.

4. The plain reasonning to go through in such a case of incertitude as to which of “malady” or “se” is the subject is to ask oneself who or what is doing the action indicated by the verb or in a second and only other domain of possibilities who or what is declared by the verb to be in a given state. Here “se trouver” can only be associated with the latter case as this verb means “to be found”¹; it is clear that a verb that indicates a state cannot have an object and so “se” is not an object; we are not dealing with the typical pronominal verb constructed with a reflexive personal pronoun but with a verbal locution in which “se” does not function as it does habitually.

   ¹ _{If we think of this translation we might understand what apparently confuses you when you mingle active with passive form: “to be found” is a passive form in English, isn’t it (The path was found by a soldier.); however that is not the English verb needed; the verb needed is the verb in the following example: “The mountains are found further to the East.”; that one is not a verb in the passive form, as you will agree.}