I know (or at least it seems to me that) you lose the grace period following the next month's billing cycle if you don't pay the balance of the current/most recent one in full during its grace period (assuming you paid off the last two month's balances off in full by their due date, that is), but does that mean the 25-day gap between my next month's billing cycle closing date and due date is closed up completely, meaning my due date becomes the billing cycle closing date? Also, will my interest start being charged to my account (and not just calculated) from the day of the first transaction in my new billing period (or the first transaction of the previous grace period in which I didn't pay off the full balance)?
Assuming you're in the United States, the typical consequence of not paying off your balance during a grace period is that the interest is calculated retroactively and applied to your account. In other words, during the grace period, the bank may be computing the interest charge it would normally assess but is not actually applying it due to the grace period itself.
I would caution that this is a generalization, and the best source of information for your question is to review the bank's own cardholder agreement, which must spell out all of the terms and conditions for use of its card.
This may not be a serious consequence if a)your balance during the period is low, or b)the interest rate being applied to the account is relatively low. Still, it only makes sense not to give away money you don't have to by paying off the balance during the grace period, when every penny of your payments goes toward principal. Why not make good use of this?
I agree with JoeTaxpayer that you should include a country, but just assuming you're in the US or another country that commonly uses the average daily balance method here is the concept.
Assuming your card is issued by a big reputable bank and targeting average consumers (not bad credit folks) your card probably uses the average daily balance for interest calculation.
Say your card uses a calendar month as your statement period (it probably doesn't but this makes the illustration easier), and your payment due date is always on the 25th of the month. You have a brand new card issued January 1, and from January 1 to January 31, you spend $1,000 on your card. On February 1 your statement is issued, with a due date of February 25.
You pay your $1,000 balance before February 20. No interest was charged because you have an interest grace period. No interest will be due on your February statement either because you did not carry a balance through the February 25 due date.
You pay $500 toward your $1,000 balance before February 25. No interest was charged on your February 1 statement (due the 25th) balance because your card was new and this is the first time you will be carrying a balance. On March 1 your February statement is calculated and your average daily balance from February 1 to February 28 was $900 including new charges you've made. On March 1 your interest charge of $11.25 will be added to your balance (15% APR on $900). You now have until March 25 to pay $1,261.25 ($1,250 + $11.25).
You've probably continued spending on the card. A new charge of $20 hits your account on March 2. You pay your full $1,261.25 on March 3. Because of quirks related to the literal application of your payment, a carried balance of $20 remains. When your March statement is generated on April 1 you will have an average daily balance calculated as:
March 1: $1,261.25 March 2: $1,281.25 March 3: $1,281.25 March 4: $ 20.00 March 5: $ 20.00 etc
Even though you made your payment well before the due date, you have a carried balance and you will have an interest charge on the next statement. It typically takes two months of paying the entire statement balance to completely remove any carried balance and restore a fully interest free relationship.
Once you have a carried balance, stop spending on the card.