Those problems you encountered are really happening to everyone and I could recognize myself in your photos.
I have a 3 year old and an 18 months old, and had to practice with dof.
It's very tricky to shoot kids with a small f number. You can do it with babies, because they don't move, but with playing kids and when you're so close to them, is very hard thing to pull of. They're fast with their body, head movements and they're hard to follow. Try with 4 or 5.6 when you're so close to kid or you can choose that small aperture but you'll need to back of from kid, that would give you more dof. If you back of and let him play and relax, you could shoot him and after crop the surrounding.
With shutter speed don't go under 160, you might get away with 125 but 160 is better
With those settings you mentioned, ISO is very high, you shot wide open, and with 6D, I can only presume that it was a very cloudy overcast day.
You have some nice equipment, enough for shooting indoors I think, you don't need any artificial lights (although others will disagree, my personal preference is available light )
... you do need to think how you're going to use that available light and how to make it works for you.
You can use almost anything to practice how to use your available light, with your window.
Place a doll or something and shoot that. Analyze the quality of the light that falls on the doll. Is it harsh-maybe some curtains will help, maybe the doll was too close/far away from the window? Do you and where do you need a reflector? what do you mind in the background?
That practice will help you a lot, may sound silly but it will help. You'll now where is the best spot to place a child, where would you stand, where is the best direction for him to go and then act according to that. Get to know your environment, how light is changing in the house depending on the the time of the day.
And the last and the most important thing when you shoot kids is that you have to be a fool when shooting kids!

really! if you do everything right and get nothing in their eyes or expression, then you didn't succeed. Make some funny noises, faces, anything will do just to catch a smile or some cute expression.
It's much easier with grown ups when you get to tell them what to do, it's not that easy with children, you can't tell them, you have to act in some ways in order to get them to do what you want them to do...grrr, my English, hope you understood!
I probably forget something to say, but have to go now. Good luck!