About a year ago, I prepared my concrete floors to be covered first with an overlay and then stained. My situation was a bit different, but I bet the prep needs to be about the same. You can read my blog post about the preparation we did on lauramakes.com.
For your situation, you will want to use a floor scraper to remove anything loose on the floor, like paint chips and the glue, or you won't get your paint or overlay to stick well. I found that wetting down the floor made the scraping go much faster.
You will definitely need to get the carpet glue up. I was interested in non-toxic products, so I looked into two soy based strippers. I tested Soy It, and it definitely worked for paint and is supposedly good for adhesives too.
After you get up the glue, you will need to take care of the grease stains. If you don't, it may show through your paint. The Eco Safety Products support person recommended that we use a floor buffer that you can rent to clean the floor first with orange degreaser and then water. Our grease stains didn't come completely out, but I don't think we found a heavy enough buffer to rent.
I can't really answer much about etching because I decided early on that the process was too toxic for us because we would be living in the house with our pets at the same time as we were prepping the floors.
Anyhow, I hope that helps a bit and let me know if I can elaborate on anything.
From the description and photograph, it appears that the crack is the natural result of the sort of stresses one would expect at a location with:
Drastic change in slab width to extend the slab through the door opening.
Drastic change in thermal stress where the slab transitions from interior temperatures to exterior temperatures.
Drastic change in slab stiffness. Dynamic vehicle loads are applied sequentially from the rigidly supported portion directly above the wall below to the spanning portion where they create deflection in the supporting structure.
All of which is to say that repairing the joint does not mean filling it in. There will always be a joint there. Instead, repairing the joint probably consists of keeping water out using backer rod and a flexible traffic sealant rated for wide joints...and perhaps a metal joint cover.
There's no magic in a can that it going to defy the laws of physics or change the internal stresses of a concrete slab under dynamic vehicular loads. Any repair needs to be premised on acceptance that there will be a joint across the opening where the slab narrows.
Best Answer
Repeating Answers from previous question but now revised. Answer to your revised question.
I would go with option (3) fill in the cracks
If repairs/sealing of the cracks is to be done, I would go with two types of products, liquid/self leveling for the thinner-fine cracks and a heavier product for the larger cracks. For the liquid, look at products similar to Dap 37584 product and for the larger cracks, Dap Ready-Mixed Patch.