dnd-5eforgotten-realmslore
Many D&D 5e adventure modules take place in the Sword Coast of Faerûn, such as Lost Mine of Phandelver (LMOP).
LMOP has information regarding various factions and mayorship but is light on other details.
How is the larger area of the Sword Coast governed?
For example: imagine the PCs join some Orc raiders and slaughter a town; who would come to enforce the law of the land?
Currently the Sword Coast Adventurer's Guide is the best source of 5e information on the sword coast. Unfortunately (for our purpose anyway) its focus is on political groups and mentions of the climate in a given area are rare. The island realm of Evermeet is mentioned as being, "the island paradise of the elves", and since most of its population is sun elves perhaps we can assume it's pretty nice? We also find out The Cold Lands are , "...in the cold, dry environs of the northeast", which is great but sadly not actually on the sword coast.
There are a couple of maps in the SCAG that we can use to infer about climate. The map of the coast shows areas near the sea of swords to be (mostly) grasslands, a couple of marshes as you move further out, and then the desert of Anauroch even further out. From this we could conclude that a fair bit of rain comes in off the sea and fades out before getting too far inland. The second map is of the north and shows that it is, shockingly, cold.
Alright well how about Hoard of the Dragon Queen. The adventure takes us north along the sword coast so we can look for references to the climate in the areas we pass through. The first mention of bad weather is around the Mere of Dead Men is, "...cold, muddy...", which is great news! The Mere of Dead Men is located north of waterdeed, right on the coast, and hemed in by mountains. Its waterlogged status fits our theory that rain from the sea is a dominant force (being held back by the mountains the rain would be very heavy in this area). Also since up to this point it has been nice enough to mention it being cold we can assume that south of here it's at least mild.
If you're interested in information from other editions then you are in luck! The Forgotten Realms Campaign Setting from 3rd edition has reams of geographic and climate info. A sidebar on page 78 about Waterdeep's climate info discusses its latitude, the nearby oceans moderating aspects on its weather, and the annual arrival of arctic winds. The level of detail available is great and you can easily find information the climate of any specific area you're interested in.
Sword Coast Adventurer's Guide doesn't mention dinosaurs at all. That's not decisive as it's not an exhaustive setting book, but it is the closest thing to a "pure" setting book for the Realms that's been published so far for 5e.
Xanathar's Guide to Everything gives some hints as to where druids might see dinosaurs: the plesiosaurus appears on the Coast and Underwater tables; the pteranodon, allosaurus, anklyosaurus, and triceratops on the Grassland table; pteranodon on the Mountain table. So maybe you've seen a plesiosaurus on a 4th-grade whale-watching trip, or a menagerie traveling up from Calimshan had an allosaurus with it? (The Coastal Encounters tables include pteranodons and plesiosaurus later in that book. Grassland Encounters include anklyosaurus, allosaurus, pteranodons, triceratops, and even tyrannosaurus rex.)
However, there's nothing in 5e's literature of the Sword Coast (SCAG, Tyranny of Dragons, Princes of the Apocalypse, Storm King's Thunder, and other adventures) to suggest that dinosaurs are native to that part of Faerûn.
…And you'll arrive in Chult, and Chult is teeming with dinosaurs. You can't spit without hitting one! Really, just take a look at this Tomb of Annihilation promotional piece that depicts Chult's Port Nyanzaru:
Dinos everywhere!
Best Answer
This answer is informed by two things1:
1. The Sword Coast Adventurer's Guide (SCAG) (pages 7-70)
2. The baked-into-the-genre-tropes modeled on social-political habits in human history
The Sword Coast as a whole isn't governed as "The Sword Coast"
The region's structure (such structure as exists) is that of local centers of power with a lot of "wilderness / lawless land" in between. Where "the law" comes from is a rough approximation (as has been true in every edition of D&D) of societal models of the Feudal period and the primacy of city-based political power similar to the ancient Greek City state model (or medieval Venice), supplemented by a few extra-national organizations - like the Lord's Alliance - that resemble coalitions like the Hanseatic League of medieval times along Northern Europe's coastal frontiers.
Cities and towns: City-states with Medieval / Feudal Overtones
While further east there are Kingdoms like Cormyr, and further south there are Amn and Tethyr, the Sword Coast region is characterized by feudal / medieval / renaissance flavored urban nodes separated by long tracts of wilderness and low population density areas (towns, villages, etc). These urban centers, and local populated areas, are run by nobles, assisted by councils in some cases (Waterdeep, Neverwinter). They each are powers unto themselves: the reach of the law is limited by the reach of the local nobles' resources and their desire to enforce their will.
There is no King, and there is no Emperor. When you refer to "the law of the land" the Sword Coast doesn't have one. Each city or locale has its own modes of governance, control, and enforcement (or attempts at it). In other words, it is nothing like the world that you live in, 21st century civilization.
How is the larger area of the Sword Coast governed?
It isn't (the larger area). That's a part of the setting (SCAG, pages 7-70).
How are laws enforced & by whom?
Locally, by local authorities
How are officials appointed?
Locally. Each locale has its own method.
How are taxes levied?
Locally.
You will find mentioned in the Waterdeep Dragon Heist, Murder in Baldur's Gate, and other published adventures references to taxes paid to a local authority.
Imagine the PCs join some Orc raiders and slaughter a town; who would come to enforce the law of the land?
There isn't a hard and fast answer (DM, you make the call). The options include:
Nobody would, as it's too far away and something more pressing has the authorities in Luskan busy up to their ears (why do you think that nobles hire adventurers in the first place?)
A local posse / armed party under the authority of a local lord
A band of adventurers / mercenaries hired by a local lord, a religious leader, or a faction
Factions: if what the adventurers / PCs did was egregious enough, their actions might attract the attention of a faction or a Faerun-wide organization like the Lord's Alliance, or the Harpers. Those organizations can put the resources together to track down the adventurers / PCs and bring them to justice.
"Someone else" with an interest in that area. (The druid who lives in that forest over there? When she lays down the law, a pack of wolves comes out of the woods and scare off (or worse) bands of brigands and such).
That is all up to the DM to put into place in a given game, or by the author in a given FR novel.
Trade connects all of the cities
As with normal human models of civilization, trade between the various regions and city-states makes for both shared interests and rivalries. Trade brings with it both prosperity and potential sources of conflict. Which leads us to ...
The role of Factions as an in-world mechanism for political intrigue
Between the Harpers, the Lord's Alliance, the Zhentarim, the Emerald Enclave, and a variety of non-state organizations (the Knights Templar in Medieval Europe make for an excellent parallel) organizations with aims and goals that are not tied to a local area act to promote their own interests in multiple areas.
The Lord's Alliance: an alliance rather than "governance"
The Sword Coast Adventurer's Guide has a description of The Lord's Alliance.
As noted above, and consistent with the tropes baked into the genre and b this game since its first edition, game specific organizations similar to the Hanseatic League promote trade and "working together where there is common interest." That is as far as a "suprastate" organization of that kind can go. If the "threat" to peace and trade is large enough, the Lord's Alliance may be able to put their rivalries aside and work together to neutralize the threat.
It is worth remembering that the tropes baked into the genre - feudal / medieval forms with nobles and local nodes of power - have been an ingrained part of the game since its original edition. How that gets applied will vary by setting, will vary by location in the setting, and vary by the DM's desires to challenge the players. The Sword Coast is described as a very decentralized region in the 5e published lore: if you like, you can think of it as a re-skinned "Wild West."
1 Notes on comments before answering: