We do have indications that #1 could have happened. Doros Harishonim (vol. 3, pp. 139ff) understands Josephus (Wars 7:6:6) to be saying that after the destruction of the second Beis Hamikdash, Vespasian nationalized all property belonging to victims of the war who had left no known heirs. (He also cites Eusebius (4:8 - should be 4:6:1) as saying that the same thing happened after the failure of Bar Kochba's revolt, although the English translation there simply says that the Roman governor Rufus - presumably the same as the Turnus Rufus often mentioned as R. Akiva's nemesis - "reduced their country to a state of complete subjection.") And he explains an episode recorded in Shabbos 116a-b, concerning Rabban Gamliel and his sister Ima Shalom arguing about their inheritance before a Roman judge, in this light too: since their father Rabban Shimon ben Gamliel had been killed during the war (whether by the Romans, the Zealots, or of natural causes), they needed to get the judge's ear so that he would release the estate to them.

[In general, confiscating the property of those condemned to execution was pretty common in the ancient world, and even later. We find Achav doing so (I Kings 21), and also Yehoyakim (Vayikra Rabbah 19:6); and Alexander the Great suggested, as a "solution" for a case where two people each claimed that the other one deserved a found treasure, that he'd have executed both of them and kept the treasure himself (ibid. 27:1).]

If, then, we suppose that R. Yehoshua ben Korchah escaped (or hid) after R. Akiva's execution, then indeed Rufus or his minions may have helped themselves to his possessions.

For possibility #5 (in my comment to the question): Avos d'R. Nassan 6:2 says that R. Akiva started out his studied by going "with his son to sit before a children's teacher" - which implies that he had already been married (and either divorced or widowed) before marrying Rachel. (The alternative would be to assume that she was willing to wait a good six years after their marriage for him to follow through on his commitment to study Torah.) Toldos Tannaim V'Amoraim speculates, then, that this son was R. Yehoshua ben Korchah. A possible support for this is that Yadayim 3:5 mentions a "R. Yochanan ben Yehoshua the son of R. Akiva's father-in-law" - which, if we take that to mean that Yehoshua was the father-in-law (i.e., that "son of..." refers back to R. Yochanan), could possibly indicate that R. Akiva named his firstborn son after the child's grandfather, a common practice in those days (as it is to this day among Sephardim).

We do have indications that #1 could have happened. Doros Harishonim (vol. 3, pp. 139ff) understands Josephus (Wars 7:6:6) to be saying that after the destruction of the second Beis Hamikdash, Vespasian nationalized all property belonging to victims of the war who had left no known heirs. (He also cites Eusebius (4:8 - should be 4:6:1) as saying that the same thing happened after the failure of Bar Kochba's revolt, although the English translation there simply says that the Roman governor Rufus - presumably the same as the Turnus Rufus often mentioned as R. Akiva's nemesis - "reduced their country to a state of complete subjection.") And he explains an episode recorded in Shabbos 116a-b, concerning Rabban Gamliel and his sister Ima Shalom arguing about their inheritance before a Roman judge, in this light too: since their father Rabban Shimon ben Gamliel had been killed during the war (whether by the Romans, the Zealots, or of natural causes), they needed to get the judge's ear so that he would release the estate to them.

[In general, confiscating the property of those condemned to execution was pretty common in the ancient world, and even later. We find Achav doing so (I Kings 21), and also Yehoyakim (Vayikra Rabbah 19:6); and Alexander the Great suggested, as a "solution" for a case where two people each claimed that the other one deserved a found treasure, that he'd have executed both of them and kept the treasure himself (ibid. 27:1).]

If, then, we suppose that R. Yehoshua ben Korchah escaped (or hid) after R. Akiva's execution, then indeed Rufus or his minions may have helped themselves to his possessions.

For possibility #5 (in my comment to the question): Avos d'R. Nassan 6:2 says that R. Akiva started out his studied by going "with his son to sit before a children's teacher" - which implies that he had already been married (and either divorced or widowed) before marrying Rachel. (The alternative would be to assume that she was willing to wait a good six years after their marriage for him to follow through on his commitment to study Torah.) Toldos Tannaim V'Amoraim speculates, then, that this son was R. Yehoshua ben Korchah. A possible support for this is that Yadayim 3:5 mentions a "R. Yochanan ben Yehoshua the son of R. Akiva's father-in-law" - which, if we take that to mean that Yehoshua was the father-in-law (i.e., that "son of..." refers back to R. Yochanan), could possibly indicate that R. Akiva named his firstborn son after the child's grandfather, a common practice in those days (as it is to this day among Sephardim).