I'm assuming that you've just started running? Since this is your first 10k ever.
As you've noted - an injury is imminent. And as people in the comments have said - every person's body is different.
You are increasing your mileage very steeply and rapidly which is the bigger concern, not so much as the speed. Your body will break at some point somewhere with these 5 times per week runs. Take all the hype a bit down and don't increase the week mileage by more than 15-20%.
I get what you are experiencing, which is the "runner's high" and it can give you crazy motivation but be objective about it. As a fellow runner I have made the mistake to increase rapidly the distance per week many times and this has always led back to 2-3 weeks of resting without runs because of some injury.
You can try and switch at some point from 5 training days to 3 because the body will start failing with recovery and couldn't cope with such a rapid increase in the distance. (also with the sprints, read below)
As you want to increase your speed, you need to do speed exercises and not so much mileage. You should look around the Internet for speed training programs for 5k and 10k but in general here's what you have to do:
Sprints, SPRINTS, SPRINTS!!!
For starters you could incorporate a 10x50m sprint after your normal runs. (you should try and run at about 80% intensity during sprinting, so that you won't die and throw up at the second sprint :D)
Warm up, I mean it! Mandatory 5 mins. warm up before the start of running. Also before starting any kind of sprint run at least 1k to kickstart that body and switch to running mode - this is after the mandatory warm up.
Some days you could skip entirely the long runs (honestly, you need just one long run per week 10-12-15km.) and after the 5 min. warm up and a 1k. run just start sprinting. You can do several types of sprints at which you'll get better over time as you progress.
Firstly try 10x600m with 3 mins. rest between sprints (during the rest you could walk and breathe, don't just stay in one place) with about 80% intensity.
Then there is the 5x1km. sprints but these are a bit more advanced and you should first do the 600m ones. Again 3 mins. rest with 80% intensity or more.
Then there are the 10x100m sprints with incline, on some hill or something and you give everything you have there, go down the hill and repeat.
Recovery after sprinting is going to be really bad, you will see. So no running on the next day.
So, let's summarize what you could do in 4 days of training per week:
First day - Sprints
Second day - Some recovery run with a convenient pace with perhaps a few increases in speed (no sprinting) for about 100m on every kilometer. About 8-10kms.
Third day - Try running 5k really fast, as fast as you can, always
warm up before that and do 1k-2k jog, then rest 1-2 mins. and start
the 5k speed run.
Fourth day - Long run day, slow recovery running for about 10-12km, no speed.
Disclaimer: I've done all of these and you should notice increases in speed but don't get your hopes up, because it could take you from 6 months to more than a year to run the 10k for 40 mins. Also depends on your weight and your running form, shoes, injuries, etc.
Sources: Me - Ultramarathon runner, currently will start working on my 5k speed again, because I stopped improving it last year just when I got to the 4:00/km.