Did Rabbi Akiva create his own midrashim?

I think it is fairly clear that he did, and that he was not working from tradition. (Indeed, I think this may be the case in general.)

Here is one instance, about how the frogs multiplied in Egypt.

Midrash Rabba Shemot parasha 10:

וַתַּעַל הַצְּפַרְדֵּעַ וַתְּכַס אֶת וגו' תני רבי עקיבא אומר צפרדע אחת היתה והיא
השריצה ומלאה את ארץ מצרים אמר לו רבי אלעזר בן עזריה עקיבא מה לך אצל הגדה כלה מדברותיך ולך אצל נגעים ואהלות צפרדע אחת היתה ושרקה להן ובאו.

Shemot 7:27-28

וְאִם-מָאֵן אַתָּה, לְשַׁלֵּחַ: הִנֵּה אָנֹכִי, נֹגֵף אֶת-כָּל-גְּבוּלְךָ--בַּצְפַרְדְּעִים.
וְשָׁרַץ הַיְאֹר, צְפַרְדְּעִים, וְעָלוּ וּבָאוּ בְּבֵיתֶךָ, וּבַחֲדַר מִשְׁכָּבְךָ וְעַל-מִטָּתֶךָ; וּבְבֵית עֲבָדֶיךָ וּבְעַמֶּךָ, וּבְתַנּוּרֶיךָ וּבְמִשְׁאֲרוֹתֶיךָ

"And if thou refuse to let them go, behold, I will smite all thy borders with frogs.
And the river shall swarm with frogs, which shall go up and come into thy house, and into thy bed-chamber, and upon thy bed, and into the house of thy servants, and upon thy people, and into thine ovens, and into thy kneading-troughs."

Shemot 8:2-4:

וַיֵּט אַהֲרֹן אֶת-יָדוֹ, עַל מֵימֵי מִצְרָיִם; וַתַּעַל, הַצְּפַרְדֵּעַ, וַתְּכַס, אֶת-אֶרֶץ מִצְרָיִם.
וַיַּעֲשׂוּ-כֵן הַחַרְטֻמִּים, בְּלָטֵיהֶם; וַיַּעֲלוּ אֶת-הַצְפַרְדְּעִים, עַל-אֶרֶץ מִצְרָיִם.
וַיִּקְרָא פַרְעֹה לְמֹשֶׁה וּלְאַהֲרֹן, וַיֹּאמֶר הַעְתִּירוּ אֶל-יְהוָה, וְיָסֵר הַצְפַרְדְּעִים, מִמֶּנִּי וּמֵעַמִּי; וַאֲשַׁלְּחָה, אֶת-הָעָם, וְיִזְבְּחוּ, ה.

"And Aaron stretched out his hand over the waters of Egypt; and the frogs came up, and covered the land of Egypt.
And the magicians did in like manner with their secret arts, and brought up frogs upon the land of Egypt.
Then Pharaoh called for Moses and Aaron, and said: 'Entreat the LORD, that He take away the frogs from me, and from my people; and I will let the people go, that they may sacrifice unto the LORD.'"

'And the frog[s] came up, and (it) covered, etc'. A brayta (See Sanhedrin page 67): R Akiva says, a single frog there was and she multiplied and filled the land of Egypt.

Rabbi Elazar ben Azarya said to him, Akiva, what have you to do with Haggada? End your words and go to (the Laws) of (the impurities) of Skin Ailments and Tents. (Rather,) there was a single frog and she whistled* to them and they came.

If he were saying this from tradition, why would Rabbi Eleazar ben Azarya criticize him, and particularly criticizing him by saying "don't quit your day job"?

Rabbi Eleazar ben Azarya also criticizes him in Yerushalmi Sotah 25b:

דרש רבי עקיבה (איוב לב) ויחר אף אליהוא בן ברכאל הבוזי משפחת רם אליהוא זה בלעם בן ברכאל שבא לקלל את ישראל ובירכן (דברים כב) ולא אבה י"י אלהיך לשמוע אל בלעם הבוזי שהיתה נבואתו בזויה (במדבר כד) נופל וגלוי עינים ממשפחת רם (במדבר כג) מן ארם ינחני בלק אמר לו ר' אלעזר בן עזריה אין הוא הוא כבר כסה עליו המקום ואין לית הוא עתיד להתוכח עמך אלא אליהוא זה יצחק בן ברכאל בן שבירכו האל שנאמר (בראשית כד) ויברכהו י"י.

Rabbi Akiva darshens Elihu ber Berachel's name as a hidden reference to Bilam. Ben Berach-el, that he came to curse Israel and blessed them. Habuzi because his prophecy was bezuya. Rabbi Eleazar ben Azarya said to him that if they are one and the same, one should not reveal it because Hashem has already concealed it for him, and if they are not, then he {presumably Elihu} will contend with you in the future. And then, perhaps Rabbi Eleazar ben Azarya also speaking here, Elihu is Yitzchak, and he gives a derivation.

Perhaps it is just conflicting derashot, but if Rabbi Akiva is really just repeating a tradition he had, one would not expect this criticism. Hashem concealed it, but revealed it with the derasha, and gave this tradition. I would say that these Tannaim were engaged in creating midrash.

Despite Rashi and Tosafot on Shabbat 96b-97a, I would posit that the same is happening in the following exchange:

ת"ר מקושש זה צלפחד וכן הוא אומר (במדבר טו, לב) ויהיו בני ישראל במדבר וימצאו איש וגו' ולהלן הוא אומר (במדבר כז, ג) אבינו מת במדבר מה להלן צלפחד אף כאן צלפחד דברי ר' עקיבא אמר לו ר' יהודה בן בתירא עקיבא בין כך ובין כך אתה עתיד ליתן את הדין אם כדבריך התורה כיסתו ואתה מגלה אותו ואם לאו אתה מוציא לעז על אותו צדיק

ואלא הא גמר גזירה שוה ג"ש לא גמר אלא מהיכא הוה (במדבר יד, מד) מויעפילו הוה כיוצא בדבר אתה אומר (במדבר יב, ט) ויחר אף ה' בם וילך מלמד שאף אהרן נצטרע דברי רבי עקיבא אמר לו רבי יהודה בן בתירא עקיבא בין כך ובין כך אתה עתיד ליתן את הדין אם כדבריך התורה כסתו ואתה מגלה אותו ואם לאו אתה מוציא לעז על אותו צדיק ואלא הכתיב בם ההוא בנזיפה בעלמא

It is possible that some of this is not the brayta but rather the interjection of the setama digmara. Even so, the Sifrei in Shelach does have the alternative suggestion of those who ascended the mountain defiantly. But the "veEla ha gamar gezeira shava..." in terms of content and language does not seem like the brayta.

Rabbi Yehuda ben Beteira is criticizing Rabbi Akiva. But Rabbi Akiva has a gezeira shava. And gezeira shava implies tradition. Perhaps more so than a typical derasha, because one is not permitted to be dan one's own gezeira shava. But the gemara just tosses off that gezeira shava, as if one is able to argue with it.

Shmuel Hanagid, in his Mevo Latalmud, says that one is allowed and empowered to argue with midrash aggadah, since it is not mipi hagevurah, unlike midrash halacha.

Maharzu, in his explanation of the principles of derash, says that even though one may not create one's own gezeira shava by midrash halacha, one may do so by midrash aggadah, and (presumably) that this is what they did. Why one is able to do so? Perhaps because it is only to teach homily, perhaps because it is not as important, or perhaps just because the rules are different. I would guess that it is because a gezeira shava is often merely a mnemonic for a halacha derived by other means which have been forgotten (and I am not innovating this explanation), so it should not be used to create and derive new halacha. But in terms of aggadah, its use is different, as a primary means of discovering covered information.

That is why Rabbi Akiva was criticized even if he was right. He was creating this midrash. And it is why Rabbi Yehuda ben Betera could argue with his gezera shava, because in this case he was dan gezeira shava leAtzmo.

Rashi and Tosafot appear to assume that there is no such distiction between midrash aggada and midrash halacha in terms of gezeira shava. Rashi on the daf writes:

ואלא הא גמיר גזירה שוה - וא"כ לא כסתו התורה דהוה ליה כמפורש ומשני רבי יהודה בן בתירא לא גמרה ואין אדם דן ג"ש מעצמו:

There are two ways of reading it. Probably, Rabbi Akiva had this tradition of gezeira shava, such that it was explicit, because it was in a derasha. And Rabbi Yehuda ben Beteira happened not to have such a tradition. That is what גמר means here, to learn it via tradition. But why criticize Rabbi Akiva in this way, that he would have to give din vecheshbon, if he had such a tradition?

Alternatively (though unlikely), Rabbi Akiva darshens this. Rabbi Yehuda ben Beteira rejects it, because a person cannot create his own gezeira shava.

Tosafot appears to understand Rashi in the former way. Thus:

גזירה שוה לא גמיר. ואם תאמר וליגמר מר' עקיבא שהוא קבלה מרבו ואור"ת שהיה להם בקבלה מנין ג"ש שבכל התורה וזו היתה יתירה על החשבון לפיכך לא קיבלה:

He asks that if all that is needed is a tradition, why not accept the tradition that Rabbi Akiva had? I think the question is good and the true answer is like the Maharzu, that indeed one can be dan a gezeira shava for himself. But Tosafot answers that there was an additional tradition, of the total number of gezeira shavas, and this was above the number, such that he did not accept it.

This seems rather forced to me. As I said above, I think that Rabbi Akiva was trying his hand at aggadah, and coming up with some very novel ideas, and was often criticized for it. And that Rabbi Akiva's gezeira shava is one he came up with by himself, and in accordance with the Maharzu.