It depends on what you mean by “potential form.”
Both れる/られる and ことができる are attached to a verb and their basic meaning is “have the ability to do.” (れる/られる has very different meanings such as passive and respect, but I ignore them for the purpose of this answer.) Because they are about the ability, the subject is usually animate.
The combination of ある+られる would be あられる, but this combination is ungrammatical for a reason I do not know.
Although the meaning of れる/られる/ことができる has large overlap with the meaning of “can” in English, it does not cover the usage of “can” in “can be red” (= may be red; possibly be red).
“Possibly” can be translated as “~可能性がある” as cypher wrote in a comment, or “~かもしれない.” Therefore, “can be red” is 赤い可能性がある or 赤いかもしれない. Although 可能性がある and かもしれない are not completely the same, I am not prepared to explain the difference now.
Below I will try to explain whether each of your examples is grammatical or not and what its meaning is.
- 赤くする is “redden (something),” and 赤くできる (or 赤く出来る, although できる in this usage is usually written in hiragana) is “have the ability of redden (something).”
- 赤くなれる is “have the ability to turn red,” but I do not know when it can be used.
- 赤くなりうる is “possibly turn red.” Unlike 可能性がある and かもしれない, [得]{う}る in this meaning can be only attached to a verb.
- 赤いがなり得る is ungrammatical because 赤いがなる is ungrammatical.
- 赤いことができる is ungrammatical because ことができる can be only attached to a verb as I wrote above.
- 赤いであられる is ungrammatical for two reasons: 赤いである is ungrammatical, and the combination あられる is somehow ungrammatical as I wrote above.
赤くいられる
sounds possible to me (I don't know if that's valid or not, but searching Google you can find examples of it, e.g.雪の下でも赤くいられる実
presumably meaning "the fruit is able to be red even under the snow") I'm just taking punts here, I don't really know for sure :P赤くできる
gets the most results of the queries I tried,赤く可能
(possible to be red),赤く能力
(ability to be red), even赤く機能
(the feature of being able to be red?) also sound plausible to me.