Grid lines on graphs
On grid lines

It depends on the points you would like to make with the graph. If you're just going to show an upward or downward trend, then the grid lines are probably redundant. If you'd need refer back to a certain point of a curve, and knowing the vertical position of that point would be crucial, then grid lines can help. It's not about the graphs (or I may go so far to say even within publication culture,) it's about the points you are trying to get across. If the grid lines will get people there with less puzzling or work, then yes to grid lines. In all other occasions, then no.


  I'm being told that displaying horizontal grid lines implies a greater
  accuracy in modelling data and therefore should be absent in the case
  of this carbon sequestration modelling since it's not the results of
  measurements


This is perhaps the oddest graphical rule I have heard in the last 12 months. For most grid lines are just an extension of the tick mark on the axes. As long as you provide the tick marks on the y-axis, anyone can draw horizontal grid lines.

It is, however, not advisable to provide tick marks or grid lines finer than what your instrument or model can discern. For example, if your measurement or prediction is in the unit of meter. Then, at most I'd just put grid lines at 0.5 m increment. I wouldn't go so far to put 0.01 or even 0.1 m increments. That would imply some precision that I never had. I believe your partner author's concern may be more related to this problem. In that case, you two need to talk and make sure at least the tick marks make sense.

On caption vs. title


  I'm told that titles are a no go, captions only. (I've found a Uni
  spec online for science papers saying titles are mandatory). I'm told
  titles are rare in journals.


Yes, they are rare in my field (biomedical.) We use captions (located below the graph) most of the time. The caption usually starts with something like this:

Figure 1. The [would have been title]
[texts explaining the graph.] 

If you have a title in the illustration, it only serves to duplicate information, making it redundant ink.

Though, depending on fields and journals, the rule may differ. Check with the journal's guideline and other published work in that journal for clues.