Learn English – What does “switch off” mean exactly in this sentence

Fuel as a verb means:

Supply or power (an industrial plant, vehicle, or machine) with fuel: power stations fueled by low-grade coal.

[Oxford Online Dictionary]

Funk is a kind of music:

music that combines elements of rhythm and blues and soul music and that is characterized by a percussive vocal style, static harmonies, and a strong bass line with heavy downbeats

[Merriam-Webster]

Funk-fueled in the sentence means the alternative rock music was characterized, powered, or influenced by a funky music style.

Edit: Fuel has always been used metaphorically unless used for vehicles, planes, power generators, etc. Please refer to the 1.3 meaning of its noun definition.

I'll break the passage up into two parts to address your two questions.

I can’t be bothered to spend much time on her until she passes the five-year mark, …

The narrator is using mark here to mean

Noun. a figure registering a point or level reached or achieved

— https://www.merriam-webster.com/dictionary/mark (sense 3h)

So the “five-year mark” is a goal of having been her father's wife for five full years. Note, Séverine almost certainly doesn't know about this goal; there is no general notion of a five-year mark as anything special. Which brings us to

…, Dad’s record to date

This is an appositive, so the phrase is describing the previous phrase (“five-year mark”). Here the narrator is using record to mean

Noun. an attested top performance

— https://www.merriam-webster.com/dictionary/record (sense 3b)

and to date to mean

up to the present moment

— https://www.merriam-webster.com/dictionary/to%20date

The narrator is explaining why she chose five years in particular: because that is the length of her father's current longest marriage. But she is also commenting on her father's relatively short marriages. One would usually use the word “record” to refer to something extraordinary or impressive — e.g., the Guinness Book of World Records — but of course five years of marriage isn't particularly extraordinary.

Putting the the parts back together, the narrator is saying she can't be bothered with Séverine until she and her father have been married at least as long as his previous longest marriage. But I get the sense that this is really just sarcasm; the narrator is giving a hurtful and dismissive reason to ignore someone she doesn't really like, while also remarking on her father's serial monogamy.