It seems to me that all you need is descriptive statistics (change from last month, mean, min, max, etc and probably graphing). Using a general spreadsheet application like Excel or Calc will do the job just fine. Remember that Excel is designed to support business calculations and what you are asked to provide is exactly that! Using a dedicated statistics software for this task (in your environment) is a waste of resources. Full stop.
However, the solution may not be straight-forward to solve in Excel or any other program. In my experience there are two main reasons:
1. The request for data is unclear.
Why do they "increasingly want data on various aspects of our activities"? It could be that the data you have provided so far has not provided support to decisions. Are the questions they really want answered possible to support with the data you can provide? Meet up with the actual decision makers or at least someone who knows what the statistics are actually used for and ask them WHY they need it. Is it used to support resourcing? Is it used to describe changes? Not even a university administration creates statistics for no reason. Most likely, what they really want to know is a handful of numbers like "change from last month", "overall sum", "hours spent on teaching vs information searches".
Do this with an open mind. You will probably learn that many of the imperfections you see in the details are less important to them. When you know their true needs, suggest a package of data, graphs, free-text report or whatever is suitable. If some parts are easy to provide, be clear about that. If something is more difficult to produce, tell them that it is is possible but time-consuming and costly. Get their buy-in before you spend time on producing the output.
2. The raw data is not optimally formatted for the calculations
First of all, if raw data quality can be improved, do that first. Update forms used for feedback, ask for output in a specific format etc. Then arrange the data and calculations in Excel to make it flexible and easy to read and troubleshoot. The trick is to use structure your data and calculations in Excel in a way that is easy to follow visually and logically. In my experience it is very useful to use different tabs for data entry, data analysis and presentation.
It seems from your examples that your input will come from a variety of sources, both manually entered and output from other systems. To get it into Excel, create separate source data tabs where you can enter or paste your raw data. For each source data tab, create a "clean up and calculate" tab where you rearrange source data and make most of the calculations. If raw data is very far from optimal or calculations are complex you may want to use several tabs or even several workbooks for this. Then create presentation tabs where you present the results from calculations in a useful format.
I'm convinced you are suffering from both these problems. Attack them in numeric order and you are well on your way. And by all means, sign up for a course in advanced Excel that is suitable for your application. Best of luck!