Learn Finnish: Emme näe horisontissa lainkaan laivaa. - We do not see a ship on the horizon at all.

Because the sentence is negative (emme näe). In Finnish, negative sentences strictly require the object to be in the partitive case, because the action is not fully completed or doesn't apply to the whole object.

What is the exact meaning of lainkaan, and can I use it in positive sentences?

Lainkaan translates to "at all" and is used almost exclusively in negative sentences. It is a synonym for ollenkaan. Notice that it ends with the enclitic particle -kaan, which is strictly the negative counterpart to -kin (also/even).

Could lainkaan be placed somewhere else in this sentence?

Yes! Finnish word order is quite flexible. You could naturally say Emme näe laivaa horisontissa lainkaan or Emme näe horisontissa laivaa lainkaan. Placing it immediately before laivaa just ties the "at all" emphasis directly to the ship.

Why does the verb nähdä (to see) take the form näe?

Because it follows the negative verb emme, it must be in the connegative form. You find this by taking the present tense "minä" form (näen) and dropping the personal ending (-n), leaving just näe. The negative verb emme already tells us that "we" are the ones not seeing.