Pets are grey when they're knocked out! Remove them from your party and they'll revive within a half hour. Or, go to your stables and click the button to revive pets--costs gold though. Or use a revival draught from your inventory. Once revived and out of your party, their health will go up slowly over time. Or you can use potions, or pay someone in the town to raise their health.

The symbols will appear solid when they're "active" and you can then hover over them for what they mean. Skull and crossbones is a passed out pet; swords mean a pet's in your party; bag means your pet's on a mission; + means your pet can level up; whatever the next icon is means they have proficiency points to be spent; egg means they're recovering from breeding (cannot breed for a while); gold means they're for sale; and heart means they can be bred with.