Of course you can! Here's the list of all the "offline" apps: https://chrome.google.com/webstore/category/collection/offline_enabled?utm_source=webstore-app&utm_medium=awesome-new-tab-page. A quick search of both the "online" and "offline" apps for the term "html" (for example) yields a whole page of results for editors and tools. To connect to a remote sever, you can use the crosh shell. There are also notepad-type and ftp clients for hand coding and file transfer. In a pinch, you can connect to another machine, to access a specific software package (like Photoshop) with Chrome Remote Desktop. And then, of course you'll have all the online coding (website) apps at your disposal. (Ex: "webmail" or "web ftp" or mysql db capabilities that your web host might offer, etc.) The only issue you might face related to web design coding is screen size. (How long can you really stare at code on any laptop screen, anyway, right?) While I make quick code updates, bug fixes and post blog articles on my Chromebook, I'd prefer my powerhouse desktop and dual monitors for very large coding and design tasks.
I really wouldn't think so. It's such a simple computer. There are a limited amount of apps that are compatible with the Chromebook. So I would say that if you like to web design, the Chromebook is not the laptop for you.