Quick Question about Harbors

Does a Harbor on another continent need a direct sea route to the capital to get connected to the trade network? I have a city situated on an inland lake that can get a harbor up the fastest, which I need to connect my overseas "colonial" resources to the homeland. But I don't want to waste the hammers if it's not actually going to connect.

So far as I know, it has to be open water (i.e., ocean) for a direct connection. Since an inland lake does not connect to the ocean, it's not connectible to your colony.

Also, the open water must not be broken by fog of war.

By the way, you don't need to connect a trade route in order to be able to use the resources. I always found this to be a bit strange. In addition, you can build the harbor in a secondary city on the mainland as long as that city is connected to the capital by roads. However, I don't think this will work with the Iroquois special UA which treats forest within the cultural borders as roads (at least it didn't work in vanilla - did they fix this in G&K?).

This article explains in detail how trade routes work.
Basically, there must be a direct route from the city to the capital, which may only consist of roads/railroads and water tiles that aren't neither covered by ice nor within two hexes of an enemy ship, and any transition from water tiles to cities and road tiles (and viceversa) must contain a city with a harbour.

So harbours in inland lakes are mostly useless if you want to establish new trade routes.