How wide or how narrow should be the repertoire of a 1800 Elo player who still wants to continue to improve?

Question

Should he have one defense against 1.e4 and one defense against 1.d4, or two defenses against 1.e4 and two defenses against 1.d4?

Should he always play the exact same sub-variation of the exact same variation of the exact same defense, or should he learn multiple different variations of that defense? For example, if he plays the Caro-Kann, after 1.e4 c6 2.d4 d5 3.Nc3 dxe4 4.Nxe4, should he always exclusively play 4...Bf5 (the Classical variation), or should he sometimes also play 4...Nd7 (the Karpov/Steinitz/Smyslov/modern variation) and 4...Nf6 5.Nxf6+ exf6 (the Tartakower/Nimzowitsch/Korchnoi variation) and Nf6 5.Nxf6+ gxf6 (the Bronstein-Larsen variation)?

Too wide and he will be a jack of all trades and master of none. Too narrow and this risks limiting his rate of improvement and he will be too easy to prepare against.

The repertoires of GMs are much wider in the computer era than before, because of the fear of opponents' preparation, which is made much easier by the use of databases.

Answer 1

How wide or how narrow should be the repertoire of a 1800 Elo player who still wants to continue to improve?

Say your aim is to improve to ELO 2000. The difference between your level of 1800 and a 2000 player is more likely not opening knowledge but

1. Tactical awareness.
2. Endgame knowledge.

Therefore focus your attention on these two areas and you will improve much faster than being an expert on the openings.

Should he have one defense against 1.e4 and one defense against 1.d4, or two defenses against 1.e4 and two defenses against 1.d4?

Learn the basic ideas on several defenses to e4 and d4. Get a feel for good piece placement and pawn structure rather than long detailed forcing lines. Learn the basic types of middlegame positions and the likely endgames that flow from certain openings that you like.

First and foremost, IMHO, study rook endgames! Nothing separates the stronger from weaker players than rook end game play.

Should he always play the exact same sub-variation of the exact same variation of the exact same defense, or should he learn multiple different variations of that defense?

Study several different variations: be aware of piece and pawn placement in each variation and why it matters. Your key aim should be to get a playable position, not to try to win straight out of the opening (this may work against weak opponents but as you aspire to beat stronger opponents it may backfire: the stronger players will likely have superior tactical awareness and endgame skill and beat you from an inferior opening position!)

Develop a good sight-of-the-board skill. That is, what does a quick glance at a position tell you? What is the material balance? Whose pieces are better placed? Are the kings safe? Who controls key squares and open files?

Above all: Enjoy the game and its rich possibilities!