is it dangerous to download stuff when using a usb boot?
Generally speaking, it is not any more dangerous than downloading data while booted off of the regular HDD. Any precautions you would take while booted off of the HDD would also apply while booting off of USB.
Where does the system store my downloads?
Normally, with a USB install, either the whole system will be running off of your RAM, which means files are not downloaded to any "persistent" location (this is called Live mode), or the USB stick itself will have a persistent location/partition for files to be saved to, and modifications to be stored (Full Install mode). So, by default, files will either be downloaded to RAM, or to the USB stick. Unless you specifically tell it to send them to your HDD. Leading to your third question...
Can it override data in my hdd?
Not in the way you are probably thinking. Most USB installs will generally have the ability of accessing your HDD, especially if it is formatted in one of the standard, common file system types (ext3/4, NTFS, FAT, HFS+, etc). This is why Live installs are commonly used to troubleshoot problems with a HDD install. However, this does not mean it will indiscriminately write to the HDD without regards of the HDD contents. If you were to save any downloaded files to the HDD, these "writes" will act the same way that any normal write would, when booted from the HDD. Meaning, the file will be written to empty space, and within the existing file system in the HDD. You would notice this because you would still have to navigate the existing file system in the HDD, so if while booted to HDD you created a folder called /variousThings, when booted to USB and accessing the HDD, you would still see and be able to write the file to /variousThings.
So, in short, no, booting from USB would not damage your HDD unless you specifically were out to do that.