Electric field lines outside two objects

Before addressing your specific questions, it should be noted that electric field lines are a useful tool to provide primarily quantitative information on the nature of the electric field and its origin and not quantitative information.

Their principal properties are:

1. Arrows that show the direction of the field at a location. By convention, the direction of the field is the direction of the force that a positive charge would experience if placed in the field. Consequently, it provides information on the nature of the source of the field (positive or negative charge). But it does not provide quantitative information on the amount of charge.

2. Line density (lines per unit area) providing information on the relative strength of the field at a location relative to another location. But it does not provide quantitative information on the actual electric field strength.

3. Application of Gauss's law and a Gaussian surface. The field lines may provide information on whether the net charge at within a specific area (or volume) is positive, negative, or zero. But it does not provide information on the actual amount of negative or positive charge.

Now, regarding the specific questions:

Are A and B conductors instead of insulating bodies?

I could be wrong (and I'm open to alternative opinion), but appears they are insulating bodies.

The direction of the electric field lines associated with the two bodies indicates that charge is not evenly distributed on the bodies. Body A appears to be negatively charged on the left side and positively charged on the right side. For body B it is reversed.

In the case of conductors the electrons, which are highly mobile, would distribute themselves on the surface of the spherical bodies to minimize repulsive forces between them. In other words, the electrons would not be bunched up on one side of each body. Even if the spheres were charged by induction, once they are separated the charge on each will redistribute to be uniform. This is demonstrated in the following link: https://www.physicsclassroom.com/mmedia/estatics/itsn.cfm

Insulating bodies, on the other hand, can be individually electrostatically charged so that positive charge is on one side and negative charge on the other side with the charge remaining so arranged. That’s because the electrons acquired by the insulating body are relatively immobile, in comparison to the conducting body.

What is the charge of A? Is the charge of A negative on the left and positive on the right?

The net charge on A is zero (see discussion below on Gauss' Law)

Yes, the charge on A is negative on the left and positive on the right Given (1) above and the fact that the direction of the field lines is into A at the left and out of A at the right.

With only this picture, how do I know what the net charge of the entire system is?

You can't know quantitatively what the net charge of the system is, just whether it is positive, negative, or zero.

Gauss' law says that the net electrical flux across a closed surface equals the net charge enclosed by the closed surface divided by the electrical permittivity of the space. See Fig 1 below for a Gaussian surface enclosing both bodies A and B. The middle field line directly between the bodies does not contribute any flux. The lines just above and below the middle line enter and exit the space enclosed by the surface, so they contribute no net flux. For the remaining lines, there is a net flux of one line entering the space enclosed. This tells us that the net charge of the entire system is negative. But it does not tell us how negative.

Fig 2 shows separate Gaussian surface around body A and B. Note that for body A there is an equal number of lines crossing the surface in each direction, for a net flux of zero. According to Gauss's law this means that the net charge on A is zero, i.e., body A is electrically neutral.

Body B, on the other hand, has one more line going in than out, so it has net negative charge.

And, what is the charge density on the right side of A?

The sketch does not provide sufficient information to quantitatively determine the charge density. What it does tell us is that the negative charge density on the left side of A is greater than the positive charge density on the right side. We know this because the density of the field lines is greater on the left side than the right side, which tells us that the relative strength of the electric field is greater on the left side than the right side.

Hope this helps.