Can the epicene personal pronoun “they’ be used regardless of semantic gender of the word?

I'm not sure what you mean by "grammatical": the meaning of this word varies between lay people, experts, and other experts. (That is, even experts argue about the precise meaning of calling something "grammatical".) I don't see why the technical expert definitions would be relevant to your question, and you haven't listed any specific linguistic theory that you are going by, so I guess you might be using the lay meaning. In that case, the answer is no: it is not grammatical to freely replace he or she with they. There are sentences where she (or he) works and they does not, such as "The lioness is hunting, and soon she (*they) will capture her (*their) prey." (You could possibly use "it" and "its" here, but not "they" and "their".)

The most established use of “singular they” (in fact, the whole set of they-pronouns are used like this, and the usage may be more common in non-subject position) is in sentences where it refers back to an indefinite or nonspecific individual:

• A good student always remembers to study their notes before an exam.

  (indefinite, nonspecific)

It may also occur in sentences where it refers to a specific individual whose identity and gender are unknown:

• The thief's footprints indicate that they don't have very long legs.

  (refers to some specific but unidentified individual; "the thief" is grammatically definite, but the gender of the referent is unknown)

For some speakers, it is possible to use they to refer to a specific individual whose identity is unknown even when it is possible to infer the gender of the individual.

Similarly, some speakers may use they in certain contexts to refer to an indefinite or nonspecific individual even when the individual is described with a semantically "masculine" or "feminine" noun. There was a Language Log post discussing the example "maybe consider you shouldn't be telling a woman what they can and can't do with their own body".

Aside from the uses mentioned above, “they” may be used to refer to a specific known individual who does not identify as “he” or “she”.

I don't know whether there might be some people who would intentionally use “they” to refer to specific individuals who do identify as “he” or “she”. That usage pattern might conceivably be adopted by someone who wanted for ideological or personal reasons to avoid ever using gendered pronouns, but I don’t think it is at all common: it would be likely to confuse the listeners, it might be considered rude to the people who are being called “they”, and it is probably "ungrammatical" (depending on what definition you use).

I think that if somebody wants to be called they, you should call him or her they. Similarly, if they want to be called he (or she) you should call them he (or she). It's common courtesy, like calling somebody who doesn't like his given name Jim rather than James (not to mention calling [untypable symbol] the artist formerly known as Prince). And I expect many people would not be happy about being called they.

Most of the OP's examples are specific people that the speaker knows, and for whom also the speaker would presumably know which pronoun they prefer (father, girlfriend, teacher). In these cases, the speaker should use the person's preferred pronoun (probably he, she, he, respectively). Furthermore, it's confusing to use they for an obviously female dog, and there's no reason to since the dog doesn't have feelings.

Nowadays, it's permissible to use they for an unknown person of a specific gender (for example, "somebody left their ring in the ladies' room").

Added

Let me explain the way the grammar used to work, at least for me. Traditionally, prescriptivists said that singular they was ungrammatical. Some people never used it all. I grew up using it, and I know the rules I grew up with, but people who grew up speaking other dialects may have had different rules.

Now that non-traditionally gendered people have appropriated they, the rules are changing.

In the grammar of singular they that I grew up using, you could singular use they only for a person whose gender was unknown, and only for unnamed antecedents. So

*Leslie called; they want you to call them back,

was ungrammatical, but

Somebody named Leslie called; they want you to call them back,

was fine.

Similarly, you could have had the following conversation.

"The electrician came today."
"What did they say needed to be fixed?"
"She said that we need a new breaker panel, and it will cost six hundred dollars.

You needed "she" (or "he") in the third line because the first speaker has met the plumber, and presumably knows their gender.

So, even according to the old rules, it was ungrammatical to use they for antecedents like "that man selling balloons", "Jack", etc.

According to the new rules, it may be possible to use it for an unknown person of a known gender. See this Language Log post. But we still don't use it for a specific person of known gender unless they want to be referred to by the pronoun they.