For both HP and Dell: if the motherboard is new enough to have PCI-E, it's pretty much guaranteed the ATX connector is standard pinout. At least I haven't see one that isn't so far, and I've worked on quite a few OEM machines back in the mid-late 2000's.
That being said, the motherboard is new enough to power its CPU from the 12V rail via 4-pin 12V connector. So the 5V rating (and thus combined 5V+3.3V rating) is irrelevant. A CD and DVD drive will rarely use more than 1 Amp on the 5V rail (most of the time, it would be much less, even when the drive is in use.) Same goes for mechanical HDDs. The chipset(s) and RAM are going to be the biggest consumers on the 5V and 3.3V rails, depending on how the motherboard is designed and which chipset / Northbridge + Southbridge it has. Generally, though, these won't consume more than 20-40 Watts combined... so again, the 3.3V and 5V rail ratings are pretty much irrelevant with newer machines like this.
When it comes to replacement PSUs, just pick one that will work with your current and future needs. This will particularly depend on what graphics card / GPU you want to use with the system, as that will likely be the biggest power consumer if you end up going with a high(er)-end one. Otherwise if you stick with the current HD5570 you have, a good quality 250-300 Watt power supply will be more than enough. In fact, even with slightly more power-hungry mid-range GPUs, you would still be fine to use a quality 250-300 Watt PSU. Whatever you do, don't go on Amazon or Ebay and buy some no-name 1000 Gigawatt PSU that's lighter than a happy mean, even if it has supposedly "good" reviews. I personally like to stick with OEM PSUs, as they are built rather well. Of course, some of them do suffer from bad caps (and if I had to guess why your current PSU is dead / not working, that's very likely the issue with it.) But overall, I still very much prefer these OEM units to no-name new garbage. The OEM PSUs are also plentiful and cheap, so you can get 2-3 for the price of a much more powerful name-brand unit (actually, many of the "branded" PSUs use the same companies as the OEM PSUs, so there's not that much of a difference - i.e. Delta, LiteOn, Chicony/HiPro, FSP/Fortron, AcBel, and Channell Well, to name a few.) The alternative is to buy a decent/good "branded" PSU at a higher price and not have to worry about it, at least for a while (i.e. Seasonic, Antec, a good deal of Corsair and EVGA units, and a few Thermatake units.... among others.) As for the old PSU - don't throw it away if you have no use for it. Either sell it or give it away locally as some people (like myself) do occasionally buy broken units off of eBay to repair for fun or use as parts. Almost all of my desktops run either repaired or salvaged PSUs I got for free or cheap.