Why did Hashem harden Par'oh's heart, and how could He then continue to punish him?
Rashi, Ramban #2 (citing the Midrash) and Seforno: Knowing that Par'oh would relent out of fear, 1 and not because he acknowledged His superiority, Hashem hardened his heart, 2 in order to then perform wonders on him, so that Yisrael (and the nations of the world should witness His greatness. See Yechezkel, 38:23 - Ramban. 3
Ramban #1 (citing the Midrash): Due to the extent of the harsh slavery that Par'oh had imposed upon Yisrael, 4 he deserved that Hashem withdraw from him the power of freewill, in order to punish him. 5
Oznayim la'Torah: Had Hashem not hardened Par'oh's heart, he would have capitulated, not because he believed in Hashem, but because he would not have had the strength to resist. 6 By hardening his heart, Hashem merely removed his inability to resist, leaving him with the freewill to retract and obey Hashem's command. 7
Seforno: Which is not really Teshuvah at all.
Ramban: And in order that Par'oh should eventually do Teshuvah (See later 9:16). See also Tzefanyah, 3:5,6 and Bo, Sh'mos 10:1,2. Indeed, the first five times the Torah records that Par'oh hardened his own heart (See later, Sh'mos 3:13 & 22, 8:15 & 25 and 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 Par'oh", to conform to Hashem's prediction in Sh'mos (4:21) - in keeping with the Pasuk in Mishlei, 21:1 "Leiv Melech be'Yad Hashem ... ".
Presumably, so that others should witness Hashem's greatness.
And that of course would not have been a valid Teshuvah.
Had he done so, it would have been a genuine Teshuvah. But of course, he did not do so!
What does 'harden his heart' mean?
Seforno: It means that Hashem gave Par'oh the strength to bear the Makos without being forced to capitulate out of terror. 1
Seforno: Had he relented because he acknowledged Hashem's greatness, Hashem would not have hardened his heart.
What is the difference between Osos and Mofsim?
Rashi #1 and Seforno (both in Pasuk 9): 'Mofsim' are miracles that prove that there is a need (power - Rosh, Pasuk 9) for the Meshale'ach (the Sender); 'Osos' are signs to prove the authenticity of the Shali'ach. 1
Rashi #2: Refer to Devarim, 13:2:2:1.
&4: Refer to Devarim, 13:2:3 &4.
Oznayim la'Torah: An Os is not necessarily something that is supernatural, a Mofeis is. When a Mofeis comes at a predetermined time, it is sometimes referred to as an Os.
Seforno: Hence Par'oh, who doubted the authenticity of Hashem demanded a Mofeis, whereas in front of Yisrael, who never doubted the authenticity of Hashem, Moshe performed only Osos - See for example, Sh'mos 4:8,9.