-Is there a specific definition of spamming on wiki?
There are broad definitions ("too many posts in too little time"/"enough posts that it becomes irritating or uncomfortable"). Context is important when taking into account whether messaging is spam or not, especially in the case of users who are friends or collaborators.
What frequency of PM counts as spamming? I know everyone has their personal definition on what spamming is, but since I got membership removal as a punishment, I assume there is an official limitation for it (similar to how idea forum said an user cannot send more than two forums within 7 days).
If you notice that someone repeatedly refuses review requests from you, it is very likely you have been spamming them. Again, a good boundary is two refusals.
There is no officially designated limitation, as that would make it extremely easy for bad faith actors to repeatedly stop just short of the limit and then argue that they cannot be disciplined for not crossing the limit, even if they have bothered many users at once. Furthermore, again, depending on users' history with conversations, some readers may be more willing to receive multiple messages before telling someone to no longer contact them.
-If, in future, I got my account banned for any not-extremely-serious reason (in my opinion, extremely serious reason means intentionally causing harm and have very bad influence), may I contact an administrator and try to persuade them to revive my account?
Please refer to this section of the Site Rules regarding appeals: https://scp-wiki.wikidot.com/site-rules#toc5
If not, may I save some of the idea forum on my old account and use new account to post it?
No, this is considered ban evasion and will result in an immediate permanent ban for all accounts involved, even if the initial ban is not permanent.
-Do you think reviewers/admins on site would prefer long email (4-5 paragraphs per email) or shorter email (1-2)? I personally tend to write long emails to people for most of the time (since I believe explaining the reason why this happened/my attitude/etc. is a good way to avoid misunderstanding and to show I am politely serious about the conversation. Furthermore, getting everything included in one email would avoid making conversation too long-term).
I personally recommend writing short messages, as that tends to be the best way to convey that you respect the time the reader will need to spend reading your text.
If you feel that there may be potential for misunderstanding, append a note like "If there's anything that needs clarifying, please let me know" or "If you need more information, please let me know; I'm not sure if I included everything".
-Would my rule-breaking be announced on site openly anywhere (written on an announcement board, marked, or something similar and everyone can see it), or is removal/banning mostly in secret (no one other than me and admin team would know)?
The staff records site is here: https://05command.wikidot.com/ Cases are retained unless they contain personally identifiable information.
-Is it rude to include my emotions about certain things in PM?
It depends on the situation. If the mention of how you're feeling is relevant to the discussion and the reader can potentially assist with the matter (i.e., "I'm frustrated with writing an incident log in this story" vs. "I'm really mad at someone at work") then it's not necessarily rude to mention something like that.
For example, would you feel like I am wasting your time if I tell you I feel like a horribly selfish person for completely not awaring my PM is spamming? I know nobody here have the obligation to read about my psychological state, but I wish to make my attitude as clear as I can in a discussion. I'm really terrible when it comes to social interaction and it would be helpful if I could get some specific directions on what to do here.
Keep in mind that ultimately everyone on the site is a stranger on the internet, and most people will not have the time to personally tutor someone on how to act appropriately on the SCP wiki—site members are expected to be able to act appropriately without additional intervention. The most I can recommend here is watching how other authors interact with reviewers in the forums and/or official Discord (mostly the latter, since the conversation is real-time in Discord), and if you're unsure about part of the writing process, make a thread asking about it in the Questions Desk forum: https://scp-wiki.wikidot.com/forum/c-891087/help:questions-desk