Understanding the concept of a "Place Field" and the difference between place cells and grid cells
Yes, that's correct. Mostly, place cells fire in different places in different contexts, and they don't necessarily share the same spatial relationship in different contexts (like in your picture, the "yellow" cell comes right before a "cyan" cell; those place fields wouldn't necessarily be adjacent in a different context.


1.1. "Field" is just a term that refers to a region of space. I'd say the Merriam-Webster definition closest is #6:


  a region or space in which a given effect (such as magnetism) exists


1.2 The picture you have is probably a schematic, but what it depicts is a recording from several different cells. Each cell is assigned a color. Every time the cell fires a spike, there is a dot placed at the location in space where the animal was at the time of the spike. You let the animal run through the maze and get a record of lots of these points, and you see that they cluster together into place fields.

2&amp;3 - These questions are getting to be a bit too broad and also you are only supposed to ask one question per submission, so I'll be very brief here. Grid cells are located in an entirely different part of the brain, the entorhinal cortex (part of neocortex), although this part does project to the hippocampus where place cells are found.

Grid cells fire in regular patterns in a spatial field, like in this image:
https://en.wikipedia.org/wiki/Grid_cell#/media/File:Autocorrelation_image.jpg

They also keep their relationship of firing locations relative to each other constant across contexts. You can think of them like a nested coordinate system.

Place cells do not do this. For a given environment, they seem to be somewhat randomly arranged relative to each other. Most likely, what actually happens with place cells is that if you could record from all of them, there would be a unique combination of cells active at any location, and this location could then be associated with other information like "I found food here."

Perhaps you could think about grid cells like a GPS location, but place cells are like a map marker.