A bank can make an infinite amount of loans. They are not constrained by anything.

They are constrained by capital and/or reserve requirements

  The way a loan works is: they make a loan to someone and then "just in time", they borrow the money from the Fed to cover that loan.

Nope. Banks can borrow against their government securities from Central Banks by "rediscounting" during discount windows, if they need extra liquidity. Note that this is not the preferred method since Central Banks usually charges a premium. These are secured loans.

See https://www.federalreserve.gov/monetarypolicy/discountrate.h...