Replace it or risk starting a fire in house. You will need to change flue pipe as well.

The TL;DR answer to your questions is NO. You simply do not have the space you need to do what you want. If you won't consider have a propane tank installed, you'll have to stick with wood, so I suggest you find a freestanding stove that you like the looks of, maybe something cool like this: http://jotul.com/us/products/wood-stoves/jotul-f-370 (I would have posted a photo, but Houzz is having some technical issues with uploading photos)

We just purchased a cape cod style house. we would like to add a wood burning fireplace in our main floor with a wood burning stove below it in the basement. I know the venting for such is possible because this is the way my parents is set up. But I am wondering if this is possible on a pre-exsisting cape cod, as well as ticket price. The area in which he would want the vent to run through shouldn’t run up the bedroom above rather to the side. Cost is not exactly a problem but I would still like to know the cheapest route to get this but still have all components we want.

if you think you might sell the dressers, you can do a temp fix with peel-and-stick wallaper (which is thicker / easier to work with than shelf paper). I think your dressers are Ikea Malm? those have such nice smooth surfaces and clean edges, perfect for wallpaper. it will be a temp solution though (the edges might start to peel where you pull open the drawers). BUT -- maybe let that be a project for later? once you get your whole room pulled together, your dresser might look fine as-is until you have time / money to replace it with a new one.