All of the various parts of the Base Game and Expansions are able to work together yes. Though I probably wouldn't play with everything under the sun, you can certainly pick and choose for variety and to customize your options.
Any of the 9 Cases work with the Investigation Deck.
You have 3 different Items cards to choose from, 2 of which use the Sidekick (Expansion 1) and Girl Friday (Expansion 2) dice. Each Item card gives you different actions.
The Sidekicks cards (Expansion 1) act as assistants in the game and you need to choose when to use their day/night abilities.
The Double Cross cards (Expansion 1) are shuffled into the Investigate deck and act as events.
The Henchmen, Gun Molls and Traps cards (Expansion 2) have a push your luck element that can offer permanent rewards like dice rolls and persistent Twist markers. These cards work in conjunction with failures during Phase 2.
Mastermind Criminals (Expansion 1) are harder to defeat, they have an extra 6th icon in their Storyline task.
The Society of Corpses Criminals (Expansion 2) are connected to icons shown on the Detective and Police Inspector cards and not only offer some new twists but they can be used to replace ‘The Spyder’ in the base game or as a stand alone set of Criminals.
And you can, of course, play with any Detective (and when playing solo most of the Police Inspectors can be chosen by the Player as well) from the Base Game or either of the Expansions.