I noticed that the interest rates on savings accounts are significantly higher than those of checking accounts. Why is this the case?
Balances in savings accounts tend to be maintained for much longer periods of time (both by choice, and by restrictions on transactions), so the bank doesn't need to keep as large a reserve and can loan out more of the money, for example as mortgages.
What stops me from keeping all my money in a savings account then transferring a monthly allowance to my checking account after interest is generated at the beginning of every month?
Nothing. Keeping your checking account balance to just a bit more than your planned spending is a good idea for a number of reasons.
And why don't banks allow for debit cards to be connected directly to savings accounts, they seem like basically the same thing from the bank's point of view.
First, that would be pretty useless, since debit card transactions would be restricted under Regulation D (limit of 6 withdrawals from a savings account per statement cycle).
Second, that would put your entire savings account balance at risk to debit card fraud, instead of the (typically much smaller) amount in your checking account.
When you choose a checking and savings account, look for a bank that offers both with no or very low fee for overdraft protection (automatic transfer from savings to checking to prevent the checking account from going negative which is called "overdraft")
It is worthwhile to note that there are some "premium checking" accounts, with velocity requirements (you must make a certain number of transactions per month), that actually pay even more interest than savings accounts. If you can meet the velocity requirement, these are better than savings accounts (higher interest, no limit on number of transactions) except that the exposure to fraud is also higher.