Credit card companies report you as being delinquent to the three major bureaus after you have been late on your payment for 30 days or more.  In your case, you have been late for 67 days, so your credit score will have already been affected.  Your credit card issuer reports to the three major credit bureaus every thrity days, so it is very important that you make a payment immediately to avoid being reported as 90 days delinquent, at which point your credit score will take another hit.  The effect that your late status will have on your credit score will be marginal and can be repaired, if you practice responsible behavior for the next 12 months.
However, you should check your next credit card statement carefully or call your credit card company now, because your interest rate very well may have gone up (perhaps considerably) due to your late status.  Being this late on your credit card payment could have a costly effect on your wallet, depending on if or how much your interest rate has increased.  If it has skyrocketed, you should try to pay your balance off in full.  If you can’t do that, you may want to consider a balance transfer.

There are no credit cards that you can apply for with a co-signer.  All credit card offers, terms and interest rates are based on the credit history of the individual.  Credit cards are not like auto loans or home loans, in which cases, co-signers are allowed.  If you are unable to get an unsecured line of credit without a cosigner, then you should apply for a secured credit card, where a security deposit serves a collateral against your loan in the place of a co-signer.
On the other hand, if by co-signer, you meant authorized user, then yes, this is something that you can do with a credit card.  An authorized user on a credit card is someone who has access to usage on the account, but is not responsible for the bill.  The primary account holder is fully responsible for the bill, however, the activity on the card will show up on the authorized user’s credit report.

You can pay back the cash advance you take against your credit limit over time, just as you would any other purchase you make with your credit card.  Any cash advance you take will be added to your existing balance.  However, this is a very expensive way to borrow money, unless you plan to pay it back immediately.
This is because there is almost always a high fee associated with cash advances on credit cards, and your credit card company will begin assessing interest on the cash advance immediately, usually at a rate above 20 percent.  Unless you are sure you can afford the finance charges that are associated with cash advances, you should probably try to find another way to come up with the cash you need.

No.  An expired credit card is just like an expired library card.  You simply get a new one at the time of expiration.  Your credit card issuer should send you a new card in the mail automatically.  If you aren’t sure that this will happen call to find out at least a few weeks before the expiration date on your card.  Once you get your new card, your credit line, balance and interest rate should all be the same, unless you’ve been notified separately of changes in those areas.

You should contact your credit card or debit card issuer immediately and report the issue.  By law, consumer liability for debit and credit card charges is limited to $50 when fraud is reported.  However, VISA and MasterCard, who control 100 percent of the U.S. debit card market and the majority of the credit card market, have gone one step further, and require that all of their credit card issuers like Capital One, Chase, Citibank, etc. adhere to a zero percent liability policy for their customers.  They also require that immediate refunds be granted on disputed charges.  Most other major credit card networks have applied these rules as well.

This means that you should be able to get your money back right away, including over draft fees that were the result of the disputed charge.  And if it turns out the merchant admits to an error after your bank has investigated the matter, that will certainly make things easier.

You can always use whatever available balance you have on your credit card.  So yes, as long as you make the minimum payment and there is a remaining open balance you can use it.  However in this situation you are being very risky for two reasons.  First of all, if you have maxed out your $500 dollar line of credit and only paid $50 dollars on the balance, the interest on the remaining $450 will likely put you close to your limit.  This means that in actuality, you would not have $50 left to spend and trying to make a transaction for that amount may result in your card being declined or you going over your limit.  Secondly, and most importantly, it’s never a good idea to max out your credit cards unless you plan to pay back a large part, or all of the balance right away.  Potential lenders like to see that you are using your credit modestly and that you are not up to your neck in credit card debt across all of your accounts.

Yes.  This is called a balance transfer, and by doing this you can transfer the credit card debt you have on one account to another credit card account with a different issuer.  Essentially the new issuer pays your original credit card company for your debt, bringing the balance on that account to zero.  Your debt then gets transferred onto a credit card account with the new issuer.  Many people transfer their credit card balances when they find a credit card with a more attractive interest rate than the one they currently have accumulated debt on.   And yes, many credit card issuers also use balance transfer offers as a marketing tool and will extend credit card offers that feature zero percent or very low interest rates on the transferred balance for a period of time, e.g. six months or a year.  Balance transfer offers sometimes include fees, so you should take that into consideration before you go this route.

It depends.  If you don’t use your credit card in a particular month, but have an outstanding balance, then yes, you will need to make a payment on your credit card account.  When you get your statement it will reflect the minimum payment due, which will include the interest assessed on your outstanding balance.  On the other hand, if the balance on your credit card account was zero at the beginning of your billing cycle and then zero at the close, you will not be required to pay anything on your account.  Additionally, even if you don’t use your credit card in a particular month, as long as you are current on your payments, your account will still report to the three major credit bureaus as being in good standing.

A credit card balance transfer is when you move the credit card debt on one account to another credit card account with a different company.  Essentially the new issuer pays your original credit card company for your debt, bringing the balance on that account to zero.  Unless you choose to close the original account, it may still be used.  Your debt then goes onto the second credit card account under that issuer’s interest rate.  Many people transfer their credit card balances when they find a credit card with a more attractive interest rate than the one they currently have accumulated debt on.  Credit card issuers also use balance transfer offers as marketing tools and will sometimes extend credit card offers that feature lower interest rates on the transferred balance for a period of time.

Your debt to credit line ratio is calculated by dividing all of your credit card debt by the sum of the credit lines extended to you across your credit card accounts.  For example, if you had 2 credit cards, and each one had a credit line of $100 and a balance of $50 then your debt-to-available-credit line ratio would be ($50+$50) / ($100+$100) = 50 percent.
The debt to credit line ratio is calculated each time a billing cycle for any of the credit cards you have closes.  If you maxed out your card and paid down your balance before the billing cycle ended, then the high balance won’t be taken into account at all.  If the balance still stands at the close of the billing cycle, it will be used to calculate your new debt to credit line ratio, but it might not make much difference depending on how much available credit you have across all of your credit cards.