Check the motherboard manual/the internet. That beep coming from your motherboard is telling you something.
Not always so. Some PSUs are better quality than others. Try it with just the bare minimum of stuff plugged in (I.e Motherboard, CPU + heatsink, 1 stick of RAM, HDD, video card) and see if it's ok then.
Or just swap the PSU from your old case to the new one :)"