It depends on how perfect you want the floors to look. I like the older floors as they have more character. Wood floors tend to have larger gaps in the winter as the cold makes the floors shrink. You may notice the gaps disappear in the summer.
You can try vacuuming without the beater bar and/or using the attachments to get in the cracks.
If it's very bothersome, however, you probably would have to replace them.

Yes, there are techniques for filling the cracks. Google a few antique floor websites. The one I remember used manila rope strands as filler. Rope expands and contracts. http://www.thisoldhouse.com/toh/video/0,,1631562,00.html

A good refinishing company will trowel fill those gaps before applying finish making the finished floor look seamless.

Check out the technique we use.

I can't see any broken planks so I can't say if you have "rot". If they are rotten, you will need to have those areas cut out and replaced. Once the new planks are put into place the floor will be sanded level (probably resanded) and then finish added to the whole floor. This sounds worse than it is. A repair will be cheaper than buying a new floor.

I can see some planks have splintered or had broken edges; is that what you are calling "broken"? If the splinters are continuing your wood may be so dry there isn't much more that can be done. Old dry wood = old dry wood. Most hardwood floors can make it 80 - 100 years. If this is original to the home (1920) it has probably served its time and can be "let go". Some of these old slated floors were actuall "subfloors" that were designed to have "flooring" laid over top. It is hard to say, by one close up photo, which you have: floor vs. subfloor.

I have old pine floors and they do shrink in the winter and expand and tighten in the summer. They were filled years ago and it became a big pain as the filler became brittle and with the expansion and shrinking began to crack and come out. Almost all of the filler is gone and they can now move freely. A very few of the boards have cupped slightly and some of those edges have broken when hit just right, but it doesn't bother me. I love the look of the worn patina of a pine floor. I find your floor beautiful and charming and would enjoy it as is.