@Catica Not quoting so the chain doesn't get any longer lol.
For Chase, if you have opened more than 5 personal cards in the past 24 months, you will not be approved for anything from them. This is a hard rule that doesn't have any workarounds beyond pre-approval offers. Don't apply to anything else from them if you have over 5 new accounts in 24 month, you will be denied 100% of the time.
CapitalOne is a bit tricky. They are inquiry sensitive (somewhat), and if they think you will not be profitable (have a high credit score), they are less likely to approve you. You can only get 1 card from the every 6 months - business or personal. There are tricks that can be used to help get approved -- basically have a high utilization on a card to drop your credit score for a month and apply when your credit score drops. I recently was able to get an approval doing this.
Citi - most of their cards families have the once in 24 month language. There are ways around this. Targeted mailers from them often do not contain the bonus restriction language. AA Platinum cards being the most common. You can get these by creating an AA account, but if AA miles aren't useful to you, this won't help you much.
Barclays is very inquiry sensitive. People who establish an account with them and keep it for a while seem to have better luck. I have only gotten a business card from them -- I have not been successful getting a personal one so I can't really help with this one.
AMEX - by far the easiest of all. This is why you want to keep them happy. They like to see spend on their cards. They have language in their fine print saying they can take the bonus back if you close your account within the first year, and they do this regularly. In addition to this, they are likely to blacklist you if you close an account before the first year is up making it very difficult to get approvals from them. This is why it is critical to close in the 13th month after the annual fee posts. It will he refunded -- they have specific language in their T&C to this effect so there is literally no reason to close it before the fee posts.
For AMEX you can get 2 credit cards in 90 days with them. You can hold a total of 5 credit cards at a time with them. You can have any amount of charge cards with them. You can apply for the same card after 90 days with them. I have probably about 15 AMEX credit and charge cards currently.
Bank of America is another odd one. They like to see an established relationship (checking and/or savings account and/or investments with Merrill Edge). Even with all of those they are inquiry sensitive. I wouldn't think much of the citizenship question. It is common when establishing bank accounts (checking/savings) to be asked about your citizenship. Not so much for credit cards, but I guess I don't see why it matters. My wife is a non-citizen and has checking, savings, and 2 credit card accounts with BoA currently. It is just establishing a risk profile to determine how much, if any, credit to extend you. A non-citizen represents a higher risk.
BoA has CC rules as well, but they generally aren't much of a factor since you will be denied for other things before you hit their limits most likely. I believe they are -- 2 in 6 months, 3 in a year, 4 in two years maximum. This is for personal cards only -- business cards is anything goes. I've only been able to get 1 personal card from BoA and I am their ideal banking customer (lots of assets with Merrill Edge).
US Bank - inquiry sensitive and helps to have an established banking relationship with them. Easier than BoA but similar.
Navy Federal Credit Union - super easy, but I'm not sure how you will get an account as a non-citizen. Can get a new card every 90 days.
I think that sums up most of it.