[I'm making this a Community wiki -- it seems better suited to attacking this puzzle piece by piece, because all edits take part in one place. I've taken some insights from other answers, especially from Joey Dionne's. Consider upvoting them.]
After much hunting, we've found the treasure. It is ...
Make a 88 days round-trip
(Astronomy) Mercury's orbit around the sun is 88 days.
(Physics) Mercury is also an element with symbol Hg.
Go North, then West, enter a country
Up and left from Hg in the Periodic table of elements is Silver.
(Geography) Argentina is an artificial name derived from the Latin name o fsilver (La Plata).
Between the sky and the sea, in the white horizon
(Vexillology) Argentina's flag has a three horizontal stripes: Light blue (sky), white (horizon) and light blue again (sea). In the centre is the sun of May.
Sum it in France, to the largest city
(Cryptography) The A-to-Z sum of sun is 54: S (19) + U (21) + N (14)
(Geography) In France, the départements are numbered. Number 54 is Meurthe-et-Moselle, whose capital and largest city is Nancy.
Name of a "pub maiden" (or something like that), go to her fathers' home
(Music) "Nancy the tavern wench" is a song by the Scottish pirate metal band Alestorm, which hails from Perth, Scotland.
In their most known body of water
(Geography) Scotland's most notorious body of water surely is Loch Ness.
Keep only the name, in an obvious code
(Communications) Loch means Lake in Scottish, so the name proper is just Ness. I'm not sure it's an "obvious" code, but in Morse, it's
-. . ... ..., that is a dash followed by eight dots.
Dash South and point toward the East from California
(Physics) Take the name in Morse as directions in the periodic system of elements: A dash means move down; a dot means move east. Start in "California", CA, that is at Calcium, Ca.
But stop a bit sooner to avoid the cops!
Moving the full 1 south and 8 east would land us as Pd (Palladium), but we want to avoid the cops (or in other words, the Police Department, PD), so we stop at Rhodium, Rh.
Under the anchor, remove the circle and circle 7 times
(Vexillology) The state flag of Rhode Island has an anchor surrouned by a circle of thirteen golden stars. But we must remove the circle under the anchor. Right under the anchor on a ribbon is the state's motto, HOPE. Remove the O to get HPE and then Caesar-shift seven times to get an animal, namely an Owl.
One animal lead to another, through the cosmic clouds
(Astronomy) The Owl nebula is a "cosmic cloud" in the constallation Ursa Major, the Great Bear.
Append his french cousin (because we need a key!)
(Language) The bear's French cousin is probably just its name in French, ours.
Finally remove xbviilps, and get your reward.
(Cryptography) We have an eight-letter key, bearours, and an eight-letter ciphertext,
xbviilps, which we need to "remove". If we decrypt that with Vigenère, which kinda removes the letter codes from each other, we get "edfjgjca".
These letters are all from the beginning of the alphabet and we can turn them into a number by substituting a with 0, b with 1, ..., j with 9, and we get 43596920. This number can be read as latitude and longitude in degrees and minutes. One of the possibilities is 43° 59' N, 69° 20' W, which leads to a town in Maine called Friendship.
Used fields according to Hint 3:
Astronomy (Owl Nebula)
Communications (Morse code)
Cryptography (AZ sum)
Geography (Loch Ness)
Language (Bear → Ours)
Physics (Mercury, Silver)
Physics (Calcium, Rhodium)
Vexillology (Sun of May)