Do you have any target time or is your goal to finish the 10 km without stopping? Interesting to know would also be whats your actual weight in kg and your height to know roughly your phsysical properties.
10 km was my distance when i was focussing on running. I started roughly at zero at the age of 17/18 without really possible to do 3 km in a row and ended it 2 1/2 years ago.
What alwayw helped me a lot that i had a training partner who motivated me to go running even though i wasnt in the mood to go running. He was much faster and had much more endurance then i had. So with my willpower if i wanted to run with him i need to go faster.
I would recommend you to train every second day, dont have big gaps in your training schedule. Increase your distance slowly (like your training schedule). Cross trainer wont substitute the running training but is very good for a warmup´and cool down.
Before each 10 km running event i used to do an overdistance run before with heavy training clothes. (Sportsmen always train heavier then they actually do their events)
Good shoes are really important you dont really want to destroy your body!
Keep us up to date how your running is going, maybe we can motivate you on bad days