Yes/no/perhaps ... Most motor operated appliances run quite well on a simpler, and much cheaper 'modified sine wave', stepped wave, inverter. Only delicate control electronics not protected by their own power supply, like electronic ballasts, have real problems. I have run several refrigerators, a freezer, and a few air conditioners off of simple modified sine-wave inverters and all ran quite well.

I carefully monitored how hot they got when running, overheating is the major problem if they don't like the waveform, but after a few hours running they all were doing fine. I worried about it in a few cases because failing to reset the thermal limiter means the unit isn't running and you need to rescue the contents of the refrigerator or freezer. Otherwise, the unit overheating isn't the end of the world. All UL refrigeration compressors have thermal regulator switches, typically a Klixon device backed up by a fusible link, in case the first fails, to disconnect the power and prevent permanent damage.

Starting, inrush, current is typically about three times the run current for most residential refrigeration units, a lot depends on what time scale we are talking about, but high efficiency motors pull a bit more before saturating and, in theory, 10X is not impossible.

Most electronic step-wave inverters can double their output for the short starting phase, essentially drawing extra power off of internal capacitors for a fraction of a second, and most refrigeration units can be fitted with a 'hard-start' kit, essentially a large capacitor, to prevent starting current issues.

You have to watch output capacities of the inverter but all I have seen have internal fuses to protect themselves so even if overloaded they don't blow up or let out the magic smoke. Keep spare fuses on hand. Practice replacing them a few times, they don't blow often but you don't want to have to learn during an emergency. Same is true of all equipment.

Many of the better manufacturers fit the start circuit of their compressors with one or more over-sized capacitors as a matter of course. This both serves to protect the unit, limit service calls, and eliminates annoyances with lights flickering when the refrigerator starts in a home with marginal wiring.

As far as efficiency goes for modern inverters even inexpensive modified sine-wave inverters only lose 10% to 15% of the power in the conversion from DC to AC.

A lot of inverter information dates back to the 60s when square-wave inverters were common, hard to find one now, and efficiencies were quite low. I have seen quotes of as little as 50% efficiency. Modern inverters are far lighter, more compact, cheaper, efficient, self-protecting, easier on the devices plugged into them.

Another bit of misinformation is that computers will only run on true sine-wave inverters. Definitely false. Most computers have switching power supplies that easily convert a rough modified sine-wave into smooth, conditioned and well regulated DC power. I have run my PCs on a cheap inverter just about every time the lights go out for a few years now.

I will add one proviso: It helps that the PC power supply is over-sized for the load the PC places on it. If the power supply was operating at close to its design limit its capacity to deal with a modified sine-wave would be compromised. Large, high-quality PC power supplies save me a lot of grief.

I'm considering building a PC around a power supply optimized for running off 12v DC to take advantage of better efficiencies. Possibly being able to switch between two power supplies for the best of both worlds. Currently, on line power, the 120v 60cycle gets converted into 12v, 5v and 3v rectified DC by the power supply. the conversion from 12v DC power, from batteries or vehicle, to the filters and conditioned array of power the PC likes is far less drastic and more efficient.

Small modified sine-wave inverters can do a lot of things the people think they can't. Even the sub-$80 700w units from the discount store will run your average refrigerator/freezer when hooked up to the battery in a car as long as the engine is running. I use one of two of those discount store units to run power tools on jobs where we just have to run them for a few minutes and have run refrigerators on them several times.

The larger 2400w modified sine-wave units have a lot more going for them but even those units are mere fraction of the cost of a small true sine wave unit.