You have two problems.
The one you've noticed is water spots remaining on the negatives when the film has been hung to dry.
Re-washing and drying may help. It may not remedy the problem if the emulsion side of the film is affected.
After the fixer bath, wash longer than you have been to get rid of ALL the fixer that has been absorbed by the emulsion. Always strive to keep all the liquids at the same temperature to avoid stressing the emulsion which has absorbed the processing solutions. The swollen emulsion is fragile.
Instead of plain water, a 30 second acid stop bath accomplishes three things: The change in pH from base to acid will put the breaks on the developer almost immediately to ensure correct development times. It will help to preserve the fixer from being diluted by the residual high pH carried into the fix from the developer. AND the change in pH will help compress the swollen emulsion to "squeeze" more of the developer out of the emulsion faster than plain water.
Washing longer allows all the fixer to migrate from inside the emulsion into the wash water. This takes time. 20 to 30 minutes is minimal. Archival wash is more like an hour unless you use Hypo (fixer) Eliminator. Ensure the wash water temperature is the same as your other solutions (20°C - 68°F optimal) and does not fluctuate. A quick dip into diluted (according to instructions - more is not better) wetting agent will lessen the possibility for drops to form on the film surface. Wash a minute or two more to remove the wetting agent and you can hang the film in a clean, dust free place to dry.
(By the way, the water spot on the first image is very much like those you find on negatives that have had insufficient washing. The tiny white spots in the larger one is a symptom I've seen before with short washes.)
Remember that the chemicals work in the emulsion not only on the surface of it.
Your second problem is the scratches on the film emulsion from squeegee-ing the film to get rid of the water.
Either let the film hang freely to let the water drain off it or make damn sure that your fingers, chamois, or blade is absolutely and perfectly clean, minimally moist, and lightly applied or the damage will be permanent and difficult to remove. (The eye can spot an unwanted scratch line faster than any other pattern I can think of right now.)
One more thing™ The total time in the fixer should be "twice the time the film takes to clear." This "time" can be determined experimentally by immersing an unprocessed chip of film in the fixer and carefully time how long it takes to become transparent. Double this time is the correct length of time for the fixer bath. Longer than this time is not necessary or helpful.