I wouldn't rip out old wood but would restain to match new. Having said that, woods differ on how they take stains but a good contractor would know to compensate for that by doing testing b4 applying, and observing/comparing in various light conditions

Ditto^ , less noticeable with darker colors. Have you checked, you might get lucky and find wood floors under carpet. But no need to tear old floors unless their is no wear layer left.

Although I agree with most of that, whether or not you should consider adding to the existing all depends on the jobsite conditions, the possibility of weaving in new matching material and the skill level of the hardwood expert hired to do the job. The flooring must match in species, grade, width and sawing characteristics. It can be done... I've done it many times and with great success.

There is definitely not any wood under the carpet. The threshold between the wood hall and the carpeted living room is at right angles to the direction of the hardwood and finished with a trim piece running the edge and then the carpet is sunk down to the level of the hardwood. The house was built in 2004 and this is how it was finished. Here is a pic. I was only actually in it once and took a bunch of photos so I'm not positive what the flooring material is. It looked to me like traditional wood floors but could be engineered. It isn't laminate though. The MLS listing says traditional wood floors. But that threshold might be hard to tie in to new material. Probably a whole lot simpler just to put in new carpet and be done with it. There is the same threshold between the kitchen and dining but at least the boards are running the other way so not all blunt ends.

based off that picture, couple of things. 1st. not oak. it looks more like Brazilian cherry.

Second thats a prefinished floor. quite possibly solid 3/4 but could be engineered too. no way to really tell based off that pic.

that threshold at the carpet is known as a header. in order to keep the flow of the floor when adding new wood would be to feather it in as glennsfc said. it can be done. but does cost more to do. you could leave the header and then just add flooring into that room without feathing it and would save you some, but lose the flow/continuity of the floor.

Finding the same product might be hard, you could probably find the same milling size/species of floor, but the color may be hard to match perfectly. So the option would be to add raw unfinished flooring, and the refinish everything. that would help get a consistant color, could even stain it lightly to help out. All this being said, should be done by a good professional, lots of things that require a lot of skill to be done properly.