Governments don't make payments at adjustable-interest rates. If sovereign borrowing rates rise, because the financial markets decide that sovereigns have a higher rate of default then they thought before, this doesn't affect outstanding debt - only the ability to get more.
It is also important to separate local governments from the sovereign federal government. The federal government is allowed to print money to cover its debts, which dramatically reduces the concern among creditors that it may default. Local governments have no such option, however; they can only address their deficit by reducing spending or raising taxes. Both options are politically unpopular (hence why most politicians do nothing and pass the buck to their successors), and may be counterproductive because the local citizens can just pack up and move if taxes are increased, or if public services deteriorate.
Edit: I forgot to mention that the most attractive way to raise government revenue is to actually attract more productive citizens to move there and increase the tax base (aka how California managed to dig itself out of its immediate financial hole). However, this is obviously very difficult to achieve in practice.
Yes, that is true. Governments typically don't pay off the principal on their loan - more specifically, as soon as the loan matures, they take out another loan.
This is not a problem if you've got some 30-year bonds maturing (As a dollar borrowed 30 years ago is trivial to pay off today), but this is a big problem if most of your debt is in the form of rolling, 1-year, or 3-year bonds.
This is still quite different from a mortgage you need to renegotiate every 5 years, or where the interest rate is pegged to prime + X%.
For anybody who has taken a look at the balance sheets of public entities, it is shocking how poorly capitalized many of these local governments actually are, and how aggressive some of their financial assumptions are (with low interest rates and 50% of their capital held in debt, a 8% assumed rate of return is extremely aggressive, and they would need financial rock-stars to achieve this at their scale). If investors are spooked, then interest rates could spike for many other government loans, and cause contagion among local government entities that would eventually require the federal government to intervene.
On the other hand, governments are perceived to be reliable borrowers, so they can receive low interest rate loans and are expected to repay their debt. There is not really a way for governments to issue equity like companies, so if governments become lackadaisical about paying their debts, then we should expect their loans to become much more expensive as well, and this could aggravate the financial distress of many high indebted public entities that are already struggling to pay the bills.