Is the mentioned method for purifying clothes from a small amount of urine and for performing istinja' after relieving oneself correct, and what is the ruling on water splashing onto clothes during istinja'?

What you do regarding the method of washing impurities and performing istinja' (washing after relieving oneself) is correct. It is sufficient to immerse the garment in water without wringing it out, because the more correct opinion is that wringing is not a condition for its purification. And the water that falls on the garment during istinja' is pure, as long as it has not changed due to the impurity of urine.