> When Congress approves an increase of the debt, I presume it's actually allowing the issuing of more treasury bonds, is that correct?
Congress approves budgets, and when the budget has a deficit it implicitly requires the issuance of debt to cover its expenses. There technically exists a “debt ceiling” by law that they created at some point in the past (I’m not familiar with the exact history) but it’s not there as a requirement and has always been raised when the debt has hit the pre existing limit.
> Does the FED have any hard/legal limit to its "printing"? Can it just print and buy stuff like mortgage-backed securities out of the blue? Or does it always require Congress approval, for example, as the executive branch needs to issue debt?
So, to start, the executive doesn’t require congressional approval to issue debt exactly, it’s more so that it’s acting on congresses allocated budget.
The fed doesn’t have any limit in place as of not in money printing, but it is supposed to follow its mandates set by congress: maintaining low unemployment and price stability. It has been granted it’s power by congress, so congress does have the power to change this at any time, but it is generally recognized that “central bank independence” is a good thing, because it’s too tempting for politicians to print their way out of budgetary deficits.