The F70 error code on a Miele G1222 dishwasher indicates that there is a water leakage issue. From the description and pictures provided, it appears that the leak is coming from the right side, near the front of the dishwasher. The grey gunk that is present on the white rectangular component in the second picture is likely caused by water and detergent residue.
In order to determine the specific cause of the leak and the necessary repairs, it is recommended to consult the dishwasher's service manual or contact a Miele-certified technician for further diagnosis and repair. It's difficult for me to give a estimate of the repair cost without more information, it may vary depending on the specific issue and the technician's hourly rate.
It is possible for a DIYer to fix the water leakage issue on a Miele G1222 dishwasher, however, it will depend on their level of experience and expertise with appliances and plumbing.
If the leak is coming from a visible and accessible area, such as a hose or valve, then it may be a relatively simple repair that a DIYer can perform with the proper tools and replacement parts. However, if the leak is coming from a more complex or hidden area, such as the pump or control module, then it may be more difficult for a DIYer to repair and would require a higher level of expertise and specialized tools.
Before attempting any repairs, it is important to consult the dishwasher's service manual and ensure that you have the proper tools, replacement parts, and safety precautions. If you are unsure about your ability to safely and effectively repair the dishwasher, it may be best to contact a certified technician for assistance.