I hope this is not a dumb question. I am going to replace both my chain and derailleur. The old chain was fine, even though it was designed for a seven speed rear sprocket with a 46 tooth front sprocket. The bike now has a five speed rear and a 52/40 front. Now the chain is in need of replacement. My derailleur also is old; the small gears are getting round. The derailleur itself is pretty banged up too.

I am asking if anyone has an opinion on what I should replace first, the derailleur or the chain.

I also have a question on the chain length. I watched on line videos on how to measure the length of the chain for installation and it seems simple. I was wondering why I could not just use the old chain as a gauge.
Any opinions would be welcome, TM.

Your current chain was sized for a single 46 tooth chainring (aka "front sprocket") and you are going to replace the crank with a double with 52/40 tooth chainrings so your chain is very likely too short. After you choose your new 5-speed freewheel (aka "rear sprockets") size your new chain to match the 52T chainring and the largest cog you have.

BTW, you can continue to use a "7-speed" chain with the 5-speed freewheel and actually you will find most of these chains marked as 6/7/8-speed.

As to the derailleur, if the pulleys are badly worn and it doesn't shift well, it should be replace too. I assume your bike has friction shifting of some sort with stem, downtube or barend shifters so almost any rear derailleur will work and an inexpensive one should be good.

+1 Your chain is very likely too short, unless it was too long when you started. So it has to go. A worn out derailleur should be dealt with as well. Easiest to do both at once.

Using old chain as a gauge? Well, if it is too short, that really won't work.

I buy 5/6 speed chain for under $5.

Thank you guys for the good advice: Taking you advice, I went to the local bike shop and brought a chain and derailleur. The process went smoothly. I did have to go back to get a new cable. I can really tell the difference in shifting. Thanks again.