Why did Hashem harden Pharaoh's heart, and how could He then continue to punish him?
Rashi, Ramban #2 (citing the Midrash) and Seforno: Knowing that Pharaoh would relent out of fear, 1 and not to acknowledge His superiority, Hashem hardened his heart, 2 in order to then perform wonders on him, so that Yisrael (and the nations of the world - See Yechezkel, 38:23 [Ramban]) should witness His greatness. 3
Ramban #1 (citing the Midrash): Due to the extent of the harsh slavery that Pharaoh had imposed upon Yisrael, 4 he deserved that Hashem withdraw from him the power of freewill, in order to punish him. 5
Which is not really Teshuvah at all (Seforno).
And in order that Pharaoh should eventually do Teshuvah (See later 9:16). See also Tzefanyah, 3:5 & 6 and Bo, Shemos 10:1 & 2. Indeed, the first five times the Torah records that Pharaoh hardened his own heart (See later, Shemos 3:13 & 22, Shemos 8:15 & 25 and Shemos 9:7) - a sure sign that he was not interested in submitting to the will of Hashem. And it is only by the last five plagues and by the Yam-Suf that the Torah writes "Vayechazek Hashem es Leiv Pharaoh", to conform to Hashem's prediction in Shemos (4:21) - in keeping with the Pasuk in Mishlei "Lev Melech be'Yad Hashem ... " [Ramban]).
Presumably, so that others should witness Hashem's greatness.
What does 'harden his heart' mean?
Seforno: It means that Hashem gave Pharaoh the strength to bear the Makos without being forced to capitulate out of terror. 1
Had he relented because he acknowledged Hashem's greatness, Hashem would not have hardened his heart (Seforno).
What is the difference between Osos and Mofsim?
Rashi and Seforno (both in Pasuk 9): Mofsim are miracles that prove that there is Tzorech (Rosh, verse 9 - power) to the Meshale'ach (the Sender); Osos to prove the authenticity of the Shali'ach. 1