[Ethereum] Am I Able To Save Remix.IDE Files To My Device

MyEtherWallet provides an offline signer for contract deployment here. You can use this to generate -and relay- your transaction.

Edit:

For the current version (v3.8.1), you need to click "Deploy contract" first. In that page it asks for the bytecode (yes, you cannot directly paste Solidity code), however, obtaining the bytecode from the source code is trivial. You can use Remix for that. Though I'm not sure if it will work with smartphones. If not, you need to find another Solidity compiler.

They're stored in the browser's Local Storage (part of the Web Storage API), which is supported by all of the main browsers. That means the files are stored on your laptop, not in the cloud.

Specifically, each .sol file you have open in Remix will be stored as a key:value pair. The key is the name of the file, the value is the contents (i.e. the Solidity code).

For example, in Chrome, if you open the Chrome Developer Tools by pressing F12 on any given page, then open the Application tab, then under Local Storage you'll find a list of the key:value pairs which are stored for that page.

For the purpose of backing up the data manually, the key:values are stored as a database under a user's profile data, for example %LocalAppData%\Google\Chrome\User Data\Default\Local Storage.

The above examples are for Chrome, and that example path is for Windows. Adjust for browser and OS accordingly.

I can access them every time I connect with my laptop but I wonder what can happen if I change laptop/browser and the like

If you change laptop you will need to move the local storage data to the new laptop. Chrome will do this for you by syncing your profile data to the Google cloud. You will have to check other browsers' behaviours.

Different browsers store local storage data in different places, so you will need to manually convert or copy between them.