building a calendar style budget/worksheet

here's what i want to create. just like a calendar on the fridge, showing, for example, the month of say sept,  an editable cashflow/budget spreadsheet. it needs the following

1. a running total (that can carry over to the next month)

2. ability to enter both income and expenses (pay and bills will vary)

3. depending on month the # of days will change. 

4. in month "X" it should show when last days of previous month and also first few days of upcoming month see example below. Asked for help with this from AI, humans are better i think...lol. I'm willing to do the work, having issues with creating and inputting formulas. not clear on how to "layer" rules etc.

5. would like to see daily balance on bottom of each cell

6. want to leave room for future abilities to categorize both expenses and income