I'm an American Muslim who doesn't speak Arabic. Can I make (Du'a), e.g. when entering the bathroom or hearing the thunder in English or must I say it in Arabic? Can I read the Qur'an in English? I have one with both English and Arabic, but I just started learning Arabic.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Arabic is the language of the Qur'an and the language of the Prophet Muhammad (Sallallahu Alaihi wa Sallam).
Allah Says (interpretation of meaning): {… that you may be (one) of the warners, In the plain Arabic language.} [26: 194-195].
Imam Shafi'e said: 'Every Muslim is obliged to learn at least the amount of Arabic language which is compulsory for him in the prayers such as reciting the Qur'an and the supplications; since uttering them in other than Arabic language is not permissible' .
Imam Ibn Taymiyah said concerning the matter of Du'a and remembrance of Allah: 'Making Du'a in a language other than Arabic language is dislikeable. It is allowed only to the one who does not have command of Arabic language, but adopting non-Arabic words as symbols is not permissible' .
Therefore, you should hasten to learn Arabic language. Making Du'a and supplicating Allah in non-Arabic out side the prayer is permissible till you learn Arabic. But prayers and reciting the Qur'an is not allowed in a language other than Arabic language.
Know that learning Surah al-Fatiha and the other short Surahs such as the Surah al-Ikhlas (112) and al-Muawwazatain (113, 114) is very easy for the one who is serious to memorize them. In a short period one may learn them in order to perform the prayers according to the right way. Learning more than the above Surah's is not compulsory but it is Sunnah for the one who is able to do so.
We say that reciting Qur'an in a language other than the Arabic language is not permissible since the Qur'an is a miracle in its words and alphabets. Whenever the words or alphabets change it is not considered as the Qur'an and the one who recites in such way is not the reciter of the Qur'an.
Finally, know that making Du'a in English language is permissible as long as you do not know Arabic but reciting the Qur'an in another language is not permissible.
Allah knows best.