Time Reports TeamWorkPM pages
To access specific time frame of project time reports pages, you can use a url variables startDate and endDate with a date format YYYYMMDD.
So to target a single day, you would use this URL (considering that the project ID is 999)
http://yoursite.teamwork.com/999/time?startDate=20140612&endDate=20140612
Recent activity request
There is a hidden feature which will allow you to filter the content of recent activity by the type of event. TeamWorkPM documentation states that you have two options: maxItems, onlyStarred. But there is also filteredItems where you can specify type of items. We use task but others should work as well. Look into the documentation for further reference.
Never trust the TeamWorkPM documentation
I mean this in a good way. It happens that their data reference pages is older then their work on API and a feature you are expecting is already built-in. Always try to parse the JSON or XML yourself and search for the thing you’re looking for.