Not sure if you want advice, reassurance or just a shoulder to cry on.
I have none of those problems. I am not a heavy user and can get a day out of my phone with wifi and 3g both on all day, and with the weather and emails syncing every 30 minutes. And I live in a low signal area, which always drains the battery more, even when I am on wifi for data.
I invested in 2 anker batteries and a desk charger unit that can charge both phone and spare battery at the same time. Total cost about £25. So I now have 3 batteries which I can keep charged, and, if I know I am away for a long time I take one or more of them with me as spares just in case.
Phone randomly turning off is usually cured in one of two ways. Either clean the battery contacts (as suggested) and pack the battery out with a bit of card to push it closer to the terminals. Or turn off GSM and just use WCDMA.
Don't know what you mean by the internet being shocking. I have no problem with mine on wifi. However on mobile networks it will depend entirely upon the strength of your signal, and whether you have 3G or not. If no 3G then forget it. If you do have 3G or "H" you have to remember that it is still nowhere near as fast as wifi. You will have to wait for the 4G roll out and a new phone for that. The other thing you can try is a different browser. I use Dolphin and can recommend it.
My phone is not slow and switches on immediately from sleep mode. So I can't really help you with that.
HTC Sensation (Giffgaff UK)