Does kimchi always have live cultures?

At least in the U.S., there is no legal requirement to list "live cultures" or whatever on food labels. Short of contacting the manufacturer, there's no way to know for certain whether or not it may contain live cultures.

Kimchi, like sauerkraut and similar cultured foods, will continue to ferment and change flavor and texture if it has live cultures. Thus, if a manufacturer wants to ensure a more stable product and longer shelf life, they may heat treat (or otherwise process) the food to destroy active bacteria before shipping the product to consumers.

On the other hand, since many people have become interested in "live culture" fermented foods in recent years, manufacturers usually go out of their way to point out when they do maintain live cultures. I'd therefore say it's likely that at least large manufacturers will advertise their "live cultures" if present. If there's no such description on the product or on the manufacturer's website, it's more likely (though not certain) that the product has been treated in some way to extend shelf life and does not contain live bacteria. For smaller and/or local producers, you probably will need to ask directly about their processing.