Yes many motherboards now have both IDE/ATA and SATA interfaces and most budget still use IDE/ATA. As was also pointed out a card or adapter can also connect an IDE/ATA drive to a motherboard with only SATA interfaces.
Being a boot drive will not matter unless you tell the computer to boot from it in the BIOS or using a boot loader. The new computer’s operating system will boot normally from it’s own HD and read the old HD as just another drive as long as it can read the file system FAT16/32 or NTFS on the old drive.
As for Dell I owned one which had problems right out of the box. They replaced it without question but from what I’ve read they still have some quality control problems. Generally if you are buying a low to low/middle end system you will get more for your money buying a brand name especially if you find something on sale.
If you want a higher end system you generally do better building your own. One advantage is you get exactly what you want. You will know how your system works making upgrades easier. Also you will avoid compatibility issues that sometimes plague brand name systems such as Dell that use proprietary parts.
As others said your best bet for a quick fix is to just get an inexpensive power supply and try that. If it doesn’t work just return it and plug your HD into another computer and burn the data to CD/DVDs. No need for an external HD adapter unless no one will let you get inside their computer.
