I'm not sure what kind of eggs you found in your pond but they were not gold fish or koi.
Soupy green is not bad for the fish, the fish love soupy green it's bad for the fish people because they can't watch the fish. I think your bio filter may have still been cycling. But do you have any shade for your pond? If not water lilies provide quick shade for the water. Fish need shade, sun also is very good for algae which will make soupy water.
Please never boil your filter pads. You've killed all the good bacteria from them. The worms in the filter pads are very good for your pond. Don't kill the worms

I'll explain the cycle to you.
Your fish produce ammonia from the gills. Ammonia is harmful to the fish. burns gills. Good bacteria in your bio filter eat the ammonia which then produces Nitrites. Nitrites are bad for your fish, they cause brown blood disease, which leads to gasping and then death. Nitrobacteria (which accumulates in your bio filter as well) eats the nitrites and turns them into nitrates. Nirtates are harmful to your fish as well. Nitrates are plant and algae food. So the plants and algae feed on them and keep the water safe for the fish. If nitrates accumulate they can cause bloody fins and weakness(lathargic fish and possibly the peachy color?) in the fish.
This is why I think your pond was cycling still, you were getting a lot of nitrates which caused the pea soup. The agae was feasting on all the nitrates. It takes 4 to 6 weeks for your pond to cycle. Once it has cycled you bio filter will take care of the waste and your water should stay clear, that is if the bio filter is doing it's job and your pond isn't over stocked.
You say you have 60 gallons? How many fish do you have? Ich is caused by stress and can also be from over stocking or poor water quality. This could lead to other problems, which may be what is going on with your koi?
Here is my suggestion. The melafix certainly isn't going to harm your fish. It does help heal wounds, I've used it myself, but it really isn't the best thing for ich. Up your salt to .3% (1 teaspoon per gallon), remember when putting salt in only up it .1% daily. With a small pond like yours you'd probably be better off just buying the pond salt (DO NOT USE TABLE SALT) at the pet store. It has the directions on the package so you have it handy. A larger pond over 1,000 gallons it's cheaper to buy water softener salt at 98.5% pure from home depot in the 20 pound bag. You'll want to mix the salt up in a bucket of pond water, make sure it's desolved before putting it in. If your water temps are around 88 degrees the ich will be gone withen three days, if the temps are cooler it could take up to 10 days. I suggest you keep your water at .3% salt for two weeks. Not only will you know the ich is gone you'll also know that your fish wont have several other bad things that can kill them. Also, if there is any way you can get more aeration to your pond this would be a good idea. Aeration helps when fish are sick.
Cleaning the bottom of your pond was a good idea. The sludge is a breader of fish disease. You want it out. Also, the ich that you have only lives about 5 days then it falls off the fish, dies at the bottom and creates hundreds of new little ichs which the fish pick up and it starts all over again. Clean pond bottom helps clear up ich.
If you have pond plants you want to make sure that the salt is not going to kill them. Hardy lilies and cat tails will not be bothered by the salt. If you have other plants they might not like the salt? You can either do a search or let me know what you have. I have a book that tells what can handle what.
You need to test your water frequently. Your cycle is starting over. Test for ammonia, nitrites, ph, and salt. The salt not only is going kill the ich but it will also help your fish defend against high ammonia. It builds up the mucos on the gills keeping them from burning in high ammonia. If you get high nitrites you'll need to do water changes to get the nitrite level down.
About the Turkey baster lol please boil it before Thanksgiving (or maybe get a new on? LOL) you know you were playing in the fish crap with it? They make siphon vacuums for aquariums, I use one to clean out my bio filter and I think it might be perfect for cleaning the bottom of your pond? Plus, you'll get your 10% water change once a week while vacuuming! I attatched a 3/4 inch peice of pvc pipe to my siphon so I would have a long handle so I wouldn't have to crouch down while cleaning the filter (I have a monster filter and it's a pain to clean).
Okay I've talked enough. I hope I haven't missed anything. Good luck
Scott