The lines are indeed visualisations to represent a vector field.
At each point in space there is a magnetic field strength and a direction for that field.
The left hand diagram is such a representation for the magnetic field around a current carrying conductor with the current coming out of the screen.
If it was correctly drawn then the length of each of the arrows should be inversely proportional to the distance from the centre.
So this diagram gives you information about magnitude and direction.
The representation that you are perhaps more familiar with is thet in the right when the tangent to a field at a point gives the direction of the magnetic field line.
To illustrate the fact that the field is stronger near the conductor the concentric circles are drawn closer to one another.
So perhaps the second diagram does not have as much information on it as the first but it is significantly easier to draw.
However there diagrams are incomplete in that the magnetic fields are actually three dimensional and then the drawing of such diagrams becomes even more difficult.
Historically the magnetic flux density was the number of field lines per unit area and that is were the term flux (= flow) comes from with magnetic flux being the total number of lines.
You will still find lots of textbooks which are in esu, emu, cgs and Gaussian units from a time when there were also magnetic poles which followed an inverse square law just like Coulomb's law for electric charges.
So going back to your queries and the statement you made that they the magnetic field lines are visualisations and so you have some degree of artic licence with them provided you follow the simple properties:
Start and finish on themselves although it is often much clearer if
you have them starting on a North pole and finishing on a South pole.
The arrow on a magnetic field line goes away from a North pole and
goes towards a South pole or follows the right hand grip rule for
Magnetic field lines are in a state of tension. That is why a North
pole attracts a South pole!
- Magnetic field lines never cross and repel each other. That is why
two North poles repel one another!
- The closer the lines are to one another the stronger is the magnetic
field (magnetic flux density).