When you measure the replacement chain, you'll cut it at the 53" (106 link) point. The other 1/4" being due to wear on the older chain which allowed the links to move apart slightly at the pins (aka"chain stretch").

Based on the wear (1/2% stretch) you should not need to also replace the freewheel, but that's always a risk. If the new chain engages poorly and skips under load, you'll need the freewheel, but as I said the odds favor you won't.

Chain tools are readily available, and there are also plenty of turorials on chain repair and replacement so you should have no issues.

BTW- while you can use any chain for 5-8 speed, you're slightly better off if you can find one marked for 5s or 6s, since these are slightly wider and will shift smoother on the 5s spacing of your freewheel (5s and 6s are the same width, while 7s and 8s are slightly narrower).