Do I have to install a game to keep it, if it is free for a limited time?

Provided you have added the game to your library while it was free, you will be free to download it whenever you like. You can also download it onto other devices too (via your steam account).

Yes, once the game is in your Steam Library you get to keep it regardless as to whether or not you installed the game on your PC.

When you clicked "Install Game", Steam registered it to your account and provided you with a game key similar to the ones you get from the Humble Bundle or when you purchase a download code from a brick and mortar store. That game key is what signifies your ownership of the game and it's virtually linked to your account now.

From personal experience, I have acquired Minion Masters and Limbo through similar promotions without installing them, and both of them are still in my library. so I assume that for this game, the same rule applies.

Note that Valve is extremely hesitant to remove games from user libraries without their consent. In recent memory, even when games have been removed from the Steam store because the developer was banned, players who owned the game could still play it. for example, people who have bought Digital Homicide games before DH was banned from the store can still download and play them.

Valve has removed games from user Steam libraries in 2 scenarios:

• Some (but not all) games that had essential external services disabled that prevent the user from meaningfully playing the game. Note that this has to be explicitly requested by the publisher of the game..
• Games where the license key has been revoked by the retailer, most frequently because the payment processing has failed (chargeback from the card owner is the most frequent occurence).