With only e.g. 2 browsers open, the fan on the laptop seems to be running at full speed.
How do I check if there is a process that is generating a lot of stress for the hardware?
Check the Task manager. Under "Processes" you can see individual CPU usage for your processes, and there is a global graph under "Performance".
If there is dust in the cooling elements, the fan will have to work harder to keep the temperature down since the air flow is obstructed. This is common with older laptops. If you have the confidence, you can disassemble it to try to clean this, but possible warranty and such are (for good reasons) out the window.