Welcome aboard arthur!

You have asked a very good question, particularly for a Toyota Corolla. :thumbsup:
This will not be a short post in answer to your question, because your Corolla's transmission takes a specially rated transmission fluid and also has special instructions on how to change it that is
unique to Toyota's. So bear with me, read this post to the end and I think this will greatly help you out.
I'm an independent AMSOIL synthetic lubricants dealer of over 27 years experience and I can tell you for a fact that no transmission fluid will last forever. Transmission fluids do need changing periodically just like any other automotive fluid does.
Transmission fluids are colored with a pinkish/reddish dye and as the fluid reaches the end of its useful life, the color will change to first a tan, then brown and then black. This provides a visual indication of the condition of the transmission fluid. Another check to perform on transmission fluid is to smell it and see if it smells burned. I am very glad you shared the pictures that you did because we can see this fluid is indeed very old and I recommend changing it before your Corolla needs a very expensive transmission rebuild. Expensive as in around $2,000 - $2,500. Call a Toyota dealers Service Department and ask them in general what a rebuilt transmission job would cost for your Corolla. Prepare to pick your jaw up off the floor. I don't feel it's wise to risk that kind of a big financial hit by not changing the transmission fluid. By the looks of that transmission fluid, I recommend changing it. Of course, if you're independent wealthy and have a ton of money falling out of your wallet, well then I guess it's not so urgent?

:icon_lol:
If you have the owners manual for your Corolla, this will provide a list of all the maintenance items for you Corolla and the recommended service intervals. If you don't have an owners manual for your car, your local Toyota dealers Parts Department can order one for you. It's a very important manual to have.
Your Corolla's automatic transmission doesn't take just any old transmission fluid. It requires a Toyota WS rated transmission fluid and most of the automatic transmission fluids I've seen on the market
don't have the WS rating, which means of course they should not be used in your Corolla's transmission, or damage can result. How can you tell if a given transmission fluid has the needed WS rating? Look on the back of the bottle and it will list all the ratings that transmission fluid is approved for.
AMSOIL offers not only one, but two Toyota WS rated high quality synthetic automatic transmission fluids for your Corolla's transmission. The first is the Signature Series Fuel-Efficient Synthetic Automatic transmission fluid below and this is the top of the line, best performing fluid -
http://www.amsoil.com/shop/find/product/atl?zo=349698
The next choice is the OE Fuel-Efficient Synthetic automatic transmission fluid, which is also an excellent ATF at a lower price -
http://www.amsoil.com/shop/find/product/otl?zo=349698
When we consider the very high cost of a transmission rebuild, anything we can do to help a transmission last longer can save a us a lot of money and industry leading AMSOIL synthetic automatic transmission fluids can do just that.
Just how good are AMSOIL synynthetic ATF's? Click on the link below to my AMSOIL Performance Tests page and look for the, "Las Vegas Taxi Cab Fleet Study", and you will see what the results of using AMSOIL synthetic ATF in heavy duty cab service were after 100,000 miles not being changed in very severe service use, the transmission is torn down and you can see the internal parts and what kind of condition they are in.
http://www.amsoil.com/performancetests.aspx?zo=349698
I will deliver the lowest price possible to you or any other Corolla Forum member.
Now, as to changing the transmission fluid in your Corolla, this is
not something that can be done at home, or by just any auto center/repair shop. Your Corolla with the special WS rated fluid requires the use of special equipment to properly change the transmission fluid or else the transmission can be under-filled, which can lead to erratic shifting and transmission failure.
continued....