Most locksmiths will do your front door and hand you a phone number when you mention a safe. I am government safe certified and I have worked on safes inside military bases and government facilities. Safes are the part of this trade I specialized in.
Yes. Almost always, and usually without damaging it. It is the most common thing people want to know before they call, and the honest answer is that a safe is a lock like any other lock. It is a better lock, built by people who were thinking about exactly this, but it is still a mechanism and mechanisms can be worked.
The real question is not whether it can be opened. It is who you let touch it. A safe opened properly goes back into service the same day. A safe attacked with a grinder by somebody guessing is scrap with your things inside it.

Nearly every safe call in the county is one of these four, and none of them is a reason to buy a new safe.

The safe that came with the house, or the one from a relative who never wrote the number down anywhere. Routine work, and most open without a mark on them.

Batteries, boards and solenoids fail years before the safe itself does. That is a repair on a part that costs a fraction of the safe, not a write off.

Worn wheel packs and broken drive cams are mechanical problems with mechanical fixes. A new combination gets set the same visit it opens.

Big ones, old cast iron ones, wall safes and the jewelry safe in a closet. Opened properly rather than cut apart with a grinder.
Anybody can buy a drill and call themselves a safe technician. Getting cleared to work on government safes means a background check and a specialty, and it is why I have been inside facilities that do not let just any tradesman through the gate. If they will hand me a safe, the one in your basement is not going to be a problem.
Almost every safe call starts the same way: nobody knows the combination, or the thing that used to open it has quit. Both are routine.
Manipulation first. A safe that gets drilled needs repair afterward, and that is your money, not mine.

This is the one job in locksmithing where the wrong person can do real damage, so a few things are worth saying out loud.
Text me a photo of the safe, plus the brand and model off the door if you can find it. Safes vary enormously, and a photo tells me in about ten seconds whether this is a quick manipulation or a real project. Then you get a number.
Almost never. Safe work happens where the safe is, which is usually the whole reason you want a mobile locksmith. Moving a loaded gun safe out of a basement is not a service anybody wants to pay for.
Yes, and you should. Any safe you did not set the code on yourself is a safe other people can open. Used safes, inherited safes and business safes with a rotating cast of employees all want a fresh combination.
Yes, and I am not being difficult. Anyone who opens safes without asking that question is a problem for every safe owner in the county, including you. A receipt, a title, the house paperwork or a serial number match is usually enough.
Send a photo and the model. You get an honest read on whether it is a quick job or a real one, before anybody drives out.
801-834-0453