Is "SharePoint Designer" needed or would it work with SharePoint Online?
Yes, it works with SharePoint Online, and it still needed.
Can one still build workflows from scratch in SharePoint Online?
Yes, you can use the SharePoint Designer to build workflows in SharePoint
Online. you can also use Microsoft Flow to create automated workflows.
And lastly is there an ability to use InfoPath 2013 at all with Office 365 or equivalent?
Yes, you can use Infopath with SharePoint Online.
Note: InfoPath 2013 and SharePoint Designer 2013 will be the last versions of those products meanwhile Microsoft will continue to
support custom workflows built with SharePoint Designer and hosted on
SharePoint Server 2016 and Office 365.
The support for InfoPath 2013 and SharePoint Designer 2013 will match
the support lifecycle for SharePoint Server 2016, running until 2026.
Regarding InfoPath alternative
Until now, Microsoft hasn't announced an explicit alternative for SharePoint Designer. however, you can use PowerApps as an alternative for Infopath in SharePoint online
- PowerApps is a service for building and using custom business apps
that connect to your data and work across the web and mobile –
without the time and expense of custom software development.
- PowerApps also introduce a logic Flow to create automated workflows between your favorite apps and services to get notifications, synchronize files, collect data, and more.
For more details check SharePoint Designer 2016 and InfoPath 2016 in SharePoint Server 2016, What’s the alternatives ?