Wow. Loads of questions you got there. These laws are all embodiments of mathematics.

• Divergence in this context is a mathematical vector operator. Think of "divergence" this way : the measure of the amount of flow out of a specified volume minus the flow into that volume. That is "amount leaving" minus "amount entering" a given volume, the resultant is your divergence.

• Think of "Electric field displacement" this way:

• Electric field represents a region induced with electric charges.
• Displacement means distance.
  Think of a these charges moving from point to point in a surface or field.
  Their movement from point to point cover distances which is "displacement". The entities doing the movement are the charges. So we say electric field displacement that is charges per unit area that would be displaced(moved) across a layer of a conductor placed in an electric field(region where you have current flowing).

• YES! flux same analogy with water example you stated. this way electric field displacement is another way of saying electric flux density both having same symbol " D".

• Conduction current density; conduction current that is currents being conducted. Density; mass per volume.
So conduction current density is the amount of conducting current flowing on a surface( which definitely has a mass and volume) in a time.

• magnetic field intensity (H) is simply the strength of that magnetic field. Meaning how intense is the field?. While "J" is conduction current density not induction field.
Induction field: think of a generator engine. The cores where you have the magnets and coils. That's an induction field.
Trust this helps @irelandscape