Those shapes are inside Graphics[]. Any Graphics[] object can be a marker, so pretty much whatever you like. The documentation shows an example using a Wolfram 'spikey' (i.e., a polyhedron).

I don't know a better way to do major and minor gridlines.

You've reached the limits of my knowledge! I don't know where the 'standard' markers live, or what format they are. But you can make open markers with the respective shapes and specifying FaceForm. Open circles would be something like

Graphics[{FaceForm[None],EdgeForm[Thickness[0.x]],Circle[]}]

Fair enough! The issue with the marker size is that you are using characters (i.e. a font), and the scaling number doesn't affect that. Here is a slightly simplifed version that does what you want.

ListLinePlot[{Reverse[dataPairs1, 2], Reverse[dataPairs2, 2]},
 Frame -> True, PlotRange -> {{0, 20}, {0, 14}},
 FrameLabel -> {Style["Vb/\!\(\*SuperscriptBox[\(cm\), \(3\)]\)", 12],
    Rotate[Style["pH", 12], -90 Degree]},
 PlotLabel -> 
  Style["Titration of weak acids with sodium hydroxide", 16, Bold, 
   FontFamily -> Default, FontColor -> Black], 
 PlotStyle -> {Red, Blue},
 PlotMarkers -> {{Graphics[Disk[]], 13}, {Graphics[Rectangle[]], 12}},
  Mesh -> All, MeshStyle -> {PointSize[0.01], Default}, 
 GridLines -> {gridLinesF[], gridLinesF[]},
 PlotLegends -> 
  Placed[
   LineLegend[{"Acetic acid", "2-chloroaniline"}, 
    LegendFunction -> (Framed[#, FrameMargins -> -2, 
        Background -> Yellow, FrameStyle -> Directive[Red]] &), 
    LegendLayout -> "Row", LegendMarkers -> Automatic, 
    LegendMarkerSize -> {{30, 13}, {30, 12}}, 
    LabelStyle -> {FontSize -> 10}], {Right, Bottom}]]

Can I ask why you are going to all this trouble to make super-fine adjustments to the plot parameters?Have you been given a style that you are required to match precisely?

The & indicates a pure function (one without an assigned name). In pure functions, the inputs are represented by # symbols (#1, #2, etc.. # is the same as #1). In this case you are defining a specific Framed[] function which will will be applied to the legend, so the # 'stands in' for the legend.

To change the legend placement and make it more precise, instead of using defined words like Center, use a formulation like this:

Placed[LineLegend[], {{1, 0}, {1, 0}}]

The first coordinates are scaled to the overall figure (bottom right), and the second coordinates are scaled to the legend itself (also bottom right). So this says to place the legend so that its bottom right corner aligns with the bottom right corner of the graphic. To move it in a bit, you might use

Placed[LineLegend[], {{0.95, 0.05}, {1, 0}}]

Experiment with the numbers and you will get the idea.

Also, it is surprising to me that RotateLabel->False doesn't do what you (and I) expect it to. It works in Graphics[] objects, but seemingly not in ListPlot[] and other functions. Maybe a bug!

However the simpler workaround is this:

FrameLabel -> {"xlabel", Rotate["label", -90 Degree]}