1. Are you pouring boiling water into the water reservoir? Your Little Guy is designed to start at room temperature using cold/room temperature water to fill the boiler.
2. When using the Illy Espresso grind are you using the espresso basket or the supermarket basket? You need to use the supermarket basket with all store bought coffee as this is usually a coarser grind.
It’s looking like your water is coming through way too quickly and not able to build up enough pressure for any cream and therefor you will also have no back pressure for your milk. With gas, you should expect your extraction to begin around the 9-10 minute mark.
Please let me know how you go with the above information and if you feel that this has solved some of your problems.