As-salamu alaikum wa rahmatullah wa barakatuh,
I will try to answer your questions to the best of my ability in shaa Allah.
If I have a feeling that I don't believe in Allah and I feel as if it is actually my thought but I say that I do believe in Allah, not in front of people but by myself, does that show faith?
Iman (faith) refers to what is in the heart (belief in Allah Subhanahu wa Ta'ala, Hus angels, His Books, His Messengers and the Last Day), but it also includes rightous deeds as in the hadith about the branches of faith: "Faith has seventy-odd branches, the highest of which is saying Laa ilaha ill-Allah (there is no god except Allah) and the least of which is removing a harmful thing from the road." So if you say that you believe in Allah, that is indeed a sign of faith.
I want to know some signs that I have iman.
If you want to know how your iman is doing (and if Allah Subhanahu wa Ta'ala is pleased with you), then look at what He allows you to do on a regular basis. Do you perform the five daily prayers? Do you fast in the month of Ramadan? Do you pay zakat? Do you give sadaqah? Do you offer voluntary 'ibadah? Do you stay away from sins? Try and grade yourself on all these questions. Is there room for improvement in any of these matters? It may give you an indication on how your iman is doing and where to start improving it, if needed.
When I get this feeling I try and get rid of it by saying that there is no other possible God and I give facts about why each religion is wrong but I still doubt?If you have actual doubts about Islam, then negating all other religions over and over again most likely isn't going to help in strengthening your iman. Instead, try to find something about Islam that makes you fall in love with Allah Subhanahu wa Ta'ala in your heart. Reading the Qur'an and learning about the 99 names of Allah Subhanahu wa Ta'ala is a good place to start. If you are already familiar with the Qur'an, then maybe try and increase your knowledge somehow by reading tafsir, learning to recite a new surah or simply perfecting an element that you haven't revised in a while.
Making duaa to Allah Subhanahu wa Ta'ala is also an important element in strengthening one's iman. Just talk to Allah Subhanahu wa Ta'ala like you would to someone you love. Ask Allah Subhanahu wa Ta'ala that He strengthens your iman, keeps you steadfast and brings you closer to Him. There is none better to ask about anything than Allah Subhanahu wa Ta'ala.
Also, someone said that as long as I don't act or speak these shirk thoughts then it wouldn't be shirk?
They may be referring to the following hadith found in al-Bukhaari (5269) and Muslim (127) that the Prophet salla Allahu 'alayhi wa sallam said, "Allah Subhanahu wa Ta'ala will forgive my ummah for whatever crosses their minds so long as they do not act upon it or speak of it." This hadith actually refers to physical actions such as zina, stealing, drinking alocohol, murder, slander and so on; where the acting then refers to the actual sin. Regarding actions of the heart such as the one you describe, this hadith is not applicable; that's because if someone were to actually desire shirk in any way, then that person has an issue with his iman.
Is this the doing of Shaytan?
There is something called compulsive waswaas, which may be what you're suffering from, as these thoughts seem to be unwanted to you. Compulsive waswaas refers to thoughts and bad ideas that come one after another to a person's mind even though he does not want them, in such a way that he cannot rid himself of them, even though he knows and is certain that they are foolish and unacceptable thoughts. They keep coming to his mind compulsively, which causes him a great deal of anguish and distress.
If a person suffering from compulsive waswaas thinks or speaks words of kufr because of his compulsive waswaas, without being aware of what he is saying or understanding what it means (or he understands what it means but he did not intend to say it and did not say it voluntarily; rather he said it under the pressure of this waswaas which forced him to say it) then he will not be punished for it, because it's beyond his control.
In summary,
You may be suffering of a dip in your iman, compulsive waswaas or a bit of both. Allah Subhanahu wa Ta'ala and you, yourself know better. Either way, both of these issues can be sorted out by increasing your 'ibadah and strengthening your relationship with Allah Subhanahu wa Ta'ala.
