Try anything you decide on a piece of similar wood first---it is difficult to remove stains. Yes, you can do what you suggested. Sanding will need to be done after each step---except the last one. However, as most woodworkers find, finishing properly can be more work than building the project. A good finish needs proper preparation(oak does not take much), proper application of stain/colorants(again, fairly simple with oak---especially if the stain is hand applied with a rag/brush) and correct application of final finish(follow the manufacturer's instructions with NO short cuts!)

Ask your floor sander if he's ever done water popping. A thin coat of water is applied to the freshly sanded wood and allowed to dry before staining. It makes the harder grain darker, more like the deeper soft grain. We then apply three coats of finish rather than two, to allow the finish to be sanded enough to remove the fuzziness the water popping produces. When we're using oil poly, we sometimes add up to 8 ounces of stain to the first or second coat of polyurethane (to make the floor darker, more uniform, or adjust the color*), buff with a maroon pad, apply the second coat, buff with a 180 screen, then a maroon pad, then apply the final coat. When applying a tinted coat, one must take care to apply evenly without drips or puddles. We have added stain to to the first and second coat in some instances. *You can either add the same stain to the poly or a different color to alter the tone. For instance if you stain with Special Walnut and want to lessen the red, you can use a few ounces of Ebony in the poly. Before doing this it's important to make a sample to determine the process. You can't accurately gauge color on a stained floor until a coat of finish is applied. Unless someone has a lot experience with this, it's best to proceed cautiously. We mostly use DuraSeal stains and oil-modified polyurethane from Lenmar and have had no problems with compatibility or adhesion using this process. We often use gloss or semi-gloss for the under coats and satin for the topcoat. Shinier oil poly has less flatting agent so using the shinier finish should increase durability. The satin topcoat hides wear, scuffs and scratches better.

A picture would be great. I don't know if you have wood floors in other rooms but they should all match. Ideally you'll want some contrast between the cabinets and floors, so if a natural finish makes the floors too close to the cabinets you might want to consider a stain.

IMO refinishing wood floors is not usually DIY and with a dog even less so. One dog hair will make you crazy when it is stuck in the finish. My advice first find out what wood that is .Then get some real help as to what finish to choose .What is the flooring in the kitchen it does not look the same and also why layed in a different direction?How are you going to handle the patch needed whare it looks like white whatever. Patches need to be woven in to look best . My advice get a pro to do this job since you have left the baseboard on to sand IMO you are not ready to DIY this . The basborads all need to be removed before sanding and refinishing .