得る never means "capable or able" but "possible only", doesn't it?
No. ~うる can express ability "to be able to do", as dictionaries I quoted above clearly show.
e.g.
〇彼ならあの石を持ち上げられる。
〇彼ならあの石を持ち上げうる。
~うる only expresses possibility when used with non-volitional verbs. (The meaning is close to ~かもしれない for this usage.) These verbs can't have the potential form ~れる/られる since it expresses ability. You've never seen あれる, unlike あり得る, right?
e.g.
×明日にでも地震は起きれる。
〇明日にでも地震は起きうる。(=起きるかもしれない)
Can I take it when あり得る simply means "possible" we say ありうる? When it means "possible to have" we say ありえる like the example below?
There is no difference between える and うる in meaning.
得ない reads えない only as far as I can see.
うる is originally from the attributive form of a classical verb う. う is 下二段活用動詞 (shimo-nidan verb), so 未然形 (equivalent to pre-nai from) is え (e.g. えず/えぬ). That's why the negative form is only えない, and うない is ungrammatical.
Incidentally I would like to ask the difference in usage between こと and もの. こと after plain verb nominalize the verb. What about adjectives? Do we have to use もの like the example below?
もの is not a nominalizer but a noun "a physical thing" there. It's also possible to interpret that it's a pronoun for 乗り物 or 地下鉄, but, anyway, こと can't be used there because こと is for abstract things. It has nothing to do with whether an adjective or a verb is used in the modifying clause.
e.g.
〇混雑した地下鉄は、初めて乗る人々にとっては、とても煩わされるものとなり得る。
cf.
×混雑した地下鉄は、初めて乗る人々にとっては、とてもややこしいこととなり得る。
〇混雑した地下鉄に乗るのは、初めて乗る人々にとっては、とてもややこしいこととなり得る。
Can we say ものになり得る instead of ものと
Yes. It's said that となる focuses more on the result of the change and になる is on the process of the change, and となる sounds more classical/formal, but they are interchangeable in most cases.
Does it make sense if I say どんなもの instead in the following sentence?
No. ものが起こる doesn't make sense there. Needless to say, there are cases where ものが起こる makes sense.
e.g.
一度地震が起こると、続けて少し規模の小さなものが起こることが多い。
(もの is a pronoun of 地震 here.)
Is でも stronger than も for the meaning of "even" in there?
Yes.
Lastly, can both こと and もの be used to refer to both physical things and abstract things?
It differs depending on the function of those words or sentence structure.