Hashem’s signs and plagues were not done to punish the Egyptians. If that were the case, the signs and plagues could have been done privately. Rather, it was so that all the Egyptians would recognize Hashem’s greatness. Therefore, a public display in their midst was warranted.
Regarding the third question, Rabbeinu Bachya writes that “you should know” refers not only to Bnei Yisrael, but also the nations of world – meaning that everyone should recognize Hashem’s true greatness. In other words, Hashem’s signs and miracles were a message to the world to see the truth of Hashem’s existence and the consequences of doing evil toward others. We see from the fact that the “you” includes the nations of the world - that Hashem’s concern is for all of His creatures. The Sefas Emes (Parshas Beshalach 5641) explains that the words “Umasbia Lechol Chai Ratzon” – he satisfies the needs of all creatures (Tehillim 145), is not limited to food and drink, but includes even the spiritual needs of all of His creations that so desire it.
This concern that Hashem has for the welfare of all of His creations can also be seen in other ways. There is a well-known expression in the Gentile world called, “What goes around comes around” or the misapplied term known as “Karma.” For example, when a driver misbehaves and attempts to cut off another driver, and if during this process, the offending driver himself ends up in an accident, people will comment, “That’s Karma!” However, what this actually is, is a manifestation of the Torah concept of Midah K’neged Midah – Hashem punishing an individual “measure for measure”. This means that Hashem punishes an individual in the same manner that the person himself has erred. Rav Michel Yehudah Lefkowitz ZT”L (in his Darchei HaChaim Vol. I p. 389) explains that Hashem punishes people in this manner as a kindness to them. By punishing people measure for measure, a Jew or a Gentile can see clearly and truthfully in what area they have failed based on how they were punished and correct their behavior accordingly.
