A rapidly growing start-up in the biotech field. Main requirements not limited to, but include - cloud sharing, interacting through comments and messages, being able to specify deadlines, estimated time interval, time-lapsed/remaining, assign multiple tasks (task dependencies), and label their priority level, and have integration with a nice group of tools/apps (google and so on).
You are describing something close to issue tracker like redmine, jira+confluence, youtrack and etc. Redmine is absolutely free, for jira you should pay, youtrack has different licenses.
Jira is actually free now for up to 10 users! https://www.atlassian.com/software/jira/pricing
Basecamp is a great product for remote teams. It is a mindset. If you're looking for a standard project management tool with lots of features, ClickUp is a great choice. It's a bit slow (especially mobile), but in terms of features, it's unbeatable.