Plenty of people refinance and end up worse off, just later. Usually because nobody ran the break-even, or because restarting a 30 year clock quietly undid the savings on the rate. So we run the math before anything else, and sometimes the math says do nothing.
Take your total cost to refinance. Divide it by what you save per month. That is how many months you have to stay in the loan before the refinance has paid for itself. If you are moving in two years and the break-even is 31 months, it does not matter how good the rate looks. That single calculation resolves most refinance decisions in about ninety seconds.
The IRRRL, the VA streamline refinance, is one of the simplest transactions in the business. Reduced documentation, usually no new appraisal, and it exists for exactly this. If you have a VA loan and rates have moved, it is worth ten minutes.
Most conventional cash-out programs go to 80% of the value. VA can go higher on some files. FHA has its own limits.
On a rate and term refinance you are replacing it, so yes. If you have a rate in the threes, a cash-out refinance is usually the wrong tool and a second lien or a HELOC is the better conversation. That is a case where refinancing would be actively bad advice.
Typically 21 to 45 days depending on the program and how quickly documents come back.
Generally 2 to 5% of the loan amount in total closing costs, some of which can sometimes be covered by lender credit in exchange for a slightly higher rate. That trade is worth modeling both ways.
Take the cost to refinance, divide it by the monthly saving, and you have your answer in about ninety seconds. And if the math says stay put, I will say stay put.