Have you tried lengthening your stitch? The stitches look a little small to me.
When you fiddle with the tension (I am assuming upper tension, right?), do you see a difference? For example, when you make the upper tension very loose you should see loops on the underside (top thread being pulled under the fabric because upper tension is too loose). If you don't see a difference in the stitch when you change the tension, chances are the tension assembly isn't right. This is an older machine, isn't it? If someone tried to dis-assemble and re-assemble the upper tension and didn't put the parts back in the correct order, the tension assembly won't work right.
The differences you are seeing in the cut size of your squares is probably due to you changing the angle of your blade in relation to the edge of the template. This is a common problem with newbies. If you experiment, you will see that how you hold the cutter affects how close the blade is to the template edge.
In the picture, it appears that one of the blocks has seam allowances open while the other piece has both seam allowances turned to the side. The typical way to match seam allowances is to press them (just finger pressing is enough) in opposite directions so they "nestle" together. You sew with the upper seam allowances facing away from you so that the presser foot "pushes" the bulk into the other seam allowance.