There are two types of effects that change a land's subtype. One adds a new subtype, the other replaces all subtypes with a single subtype. Quicksilver Fountain's effect is a replacement effect. The land stops being what it was before and becomes just an island, losing all effects and abilities printed on the card (but not abilities gained from other sources such as Abundant Growth). This is covered in the comprehensive rules (emphasis mine):
305.7.: If an effect sets a land’s subtype to one or more of the basic land types, the land no longer has its old land type. It loses all abilities generated from its rules text, its old land types, and any copy effects affecting that land, and it gains the appropriate mana ability for each new basic land type. Note that this doesn’t remove any abilities that were granted to the land by other effects. Setting a land’s subtype doesn’t add or remove any card types (such as creature) or supertypes (such as basic, legendary, and snow) the land may have. If a land gains one or more land types in addition to its own, it keeps its land types and rules text, and it gains the new land types and mana abilities.
For your specific questions.
- Yes Quicksilver Fountain's effect does affect non basic lands. They will almost become copies of the island card. They will however lack the "basic" supertype and will not be named "Island", for effects that care about it. (such as Awakened Amalgam)
- Yes, Hinterland Harbor would not be able to produce green mana if you make it into an island. Using a replacement ability like Quicksilver Fountain's, it would be a non basic land named Hinterland Harbor with the subtype island and the ability to produce only blue mana.
- Yes, Seat of the Synod would still be an artifact. Changing the subtype does not change the type or supertypes on the card, and "artifact" is a type. Seat of the Synod would behave exactly as it did without the type change, other than now having the subtype island for cards that care about it (like Islandwalk creatures)
As an addition to your questions, Quicksilver Fountain only works on lands that are not already islands, due to the targeting restriction in the card's rule text (emphasis mine):
At the beginning of each player's upkeep, that player puts a flood counter on target non-Island land he or she controls of his or her choice. That land is an Island for as long as it has a flood counter on it.
This effect is tied to the flood counters placed by Quicksilver Fountain only, since the rules text specifies "that land". The trade off is that the ability doesn't need Quicksilver Fountain to remain in play to work, if Quicksilver Fountain is destroyed the land remains an island. Flood counters placed by any other ability (such as Aquitect's Will) are not affected by the Fountain, nor do counters originally placed by the fountain but moved to another land (say Fate Transfer with two animated lands). This will matter most for the small list of non basic lands that are islands in addition to another type, like Tundra or Steam Vents, as Fountain can't force a player to lose this access to their other color.
If your goal is to mess with colors other than blue, and your deck is mono-blue since Quicksilver Fountain would cut off your access to other colors, you could look at Reality Twist. I used it in a mono-blue deck along with Eon Hub to prevent the cumulative upkeep. Unless your opponent is playing the right combination of colors it ruins their access to proper colors with their basic lands, and non-basic lands that have land types, by swapping the color a land would produce with another color. Only Boros and Golgari color combinations are unaffected.