"I am sorry, I forgot to wash the knife."
Translation:Прости, я забыл помыть нож.
Yes, it's implied.
For an excuse you can use the full version: "Прости меня за то, что ...".
Or you can say "Прости меня, что ..." or "Прости, что ...". These are colloquial.
When you say "Прости, (pause) я забыл ..." you inform that you've done something wrong, and apologize.
I think so. I'm not a native speaker. Remember, "wash" and "clean" are two different words in English as well as in Russian. Depending on context, you can sometimes substitute one for the other but they're still two different words with two separate meanings, albeit with a little overlap.
"Sorry, I forgot to wash the knife" is what many people would say in english.
Entirely possible to also say "Sorry that I forgot to wash the knife".
The extra "I am" is however redundant and left out by people is what I am getting at. Ironically leaving out "I am" is actually more accurate to how the phrase is written in russian.
But again this perhaps is not what you call "proper english"