The gelatine base is already pretty soft when it's wet and very susceptible to damage at all stages - getting it warmer just makes it a little softer. But your wash water should be the same temperature as all your other chemicals or you can cause other problems.
It sounds like you have a combination of problems. The short ones sound like friction abrasions. Did you rest the negs on anything when you cut them up? Even when dry it takes very little to scratch a neg so you should cut the negs up as they hang after drying.
The long curved ones sound as if they have been caused by wiping the water off. Do you use a squeegee or your fingers or something else? The safest way is use your fingers as you can usually feel if there is any grit there and you rinse them in the wash water first (and don't squeeze too hard).
Film is prone to scratching at all stages - dry or wet - so every unreasonable precaution has to be taken to avoid it. To do this you need to be aware of all the stages at which scratching can occur and assess your handling methods.
For 35mm:
Scratches caused in the camera - either by dust or rough spots - usually produces very long, straight scratches parallel to the edge of the film (they are called 'tram lines'). They can also be caused by dust and dirt in the cannister light trap. Clean the inside of the camera thoroughly before and after use using a soft brush and compressed air (check all the surfaces - especially the back plate - for any small imperfections. Sometimes the paint finish isn't perfect even in expensive cameras and it can get chipped too.).
Keep film cans in their plastic can until you are ready to use them. Put them back in the can when exposed until you are ready to process.
When loading on to the spiral in the darkroom do not pull the film out of the cassette but crack it open and remove the film whilst still rolled. Make sure you don't let the film come into contact with anything whilst loading - the commonest one is letting the bottom of the film go onto the floor.
Before loading make sure the spirals and tank are clean and have no chemical residues or dust in them.
Ensure that your chemicals have no suspended matter in them. Scratches can be caused by small crystals of chemical during agitation, beleve it or not. Filter the chemicals if necessary.
For the same reason, check your water supply. You would be amazed at how much grit and rust is present in ordinary tap water. You can install an expensive filter - or use a large bucket. Fill it with the water you need to make up chemicals and for the wash, get it to temperature and let it stand for 5 minutes so that any suspended particles can drop out.
Use the fill and empty method for the wash rather than force washing from the tap - for the reason mentioned above. If you have gritty water it will sandblast your negs.
Take the wet negs off the spiral only where you are going to hang them. Use a proper drying cabinet with air filters, but if you can't afford one then the best place is in a shower cubicle with the door/curtain closed.
Cut the negs into strips as they hang and put each strip straight into a neg bag. Do not use second hand neg bags.
After this only take the negs out of the bag when you have to. Do not leave them lying around uprotected. Never slide negs around on any surface, including photo paper and in the enlarger.
Keep your darkroom as clean as you can. Vacuum regularly, strip everything down and thoroughly wipe all surfaces - especially just before processing or printing.
I think that covers most danger points.
Remember: Cleanliness is next to Ansel Adams!