it seems your plants have been producing a lot of babies! If the "root-like things growing out of the junctures that also have leaves" come in contact with soil, they will take root. I have helped them along by pinning these baby plantlets onto soil with cut off paper clips, and water them in well. Once they took well, I severed them from the mother plant. In your case, they are hanging in air. You may need to put one large or several smaller pots with soil near the mother plants, and place the babies on the soil etc.
I am rather new at this myself, my 1st year experimenting with strawberry plants, and am also curious to find out why your plants decided to go into baby production and neglected fruit production? Perhaps the type of nutrition they were getting? Maybe mine were still producing some small fruit as well, because I allowed the babies to take root in soil and therefore they were drawing less energy from the mother plant?
Hope someone else on this forum will answer.

Those are runners. some varities produce alot of runners, some dont. If you want a good amount of berries, you need to continuously prune off those runners so the mother plant will put its energy into producing fruit and not runners. It is a continual maintenance required for good fruit production. after a couple of years, the plant will stop producing berries, tgen you can take the runners and start new plants with then. you should get 2 or 3 years of good fruit production from a plant. You are probably better off starting new once they stop producong rather than using runners as the runners will have any diseases that the mother plant may have picked up over the years.

If I want to turn them into more plants, can I cut them off and repot them so they dont have to stay connected? If so...where do I cut the vine (i.e. by the mother plant or close to the new baby offspring).

Also, should I replace any of my old plants with these? If so, how do I know which ones to replace?

(sorry for the questions that seem elementary but I am new at strawberries and have not had a great growing year since transporting my plants from CA to IL in July)

They don't have to stay connected to the parent plant, but it does add a measure of insurance that your efforts at propagation will be successful. Cut the main vine on both sides of the plantlet. The plantlet will have preformed adventitious roots & will take root easily in a free draining soil kept just damp.

Your lack of fruiting is very probably tied to a lack of light. SBs need full sun to fruit well.

Al