Yes, you can run IDE and SATA drives in the same machine - should not be a problem. For example, my machine has two IDE devices - both DVD rewriters - on separate IDE ports, and two SATA drives on SATA ports. It also has a floppy drive and an external drive on an eSATA port when I use it. This past weekend I added to it an old IDE HDD on one of the IDE ports.
You must understand about jumpers. In general, you must set jumpers on IDE DRIVES ONLY. (Some SATA drives have jumpers and pins on them, but they are for very different purposes.) Any IDE port and cable can support TWO devices, so each needs a unique identifier. The system used is to set each of them as either a Master or a Slave OF THIS ONE PORT ONLY. To be used at all, an IDE port MUST have a Master. IF it has a second device, that one must be the Slave. These are set by placing jumpers on pins on the back edge of the unit, and the HDD should have a diagram on its label how to do this. Please note there are NO Master or Slave jumpers to set on any SATA device (only one device per port). There is no such thing as a Master Drive for the whole machine, with a bunch of subordinate Slaves. There IS a Boot Device for the machine and that is set up in BIOS, NOT via jumpers on drives.
An IDE data ribbon cable these days has 80 wires in it, even though the connectors only have 40 (well actually, 39) sockets in them. Each cable has three connectors on it (usually) and each of these has one pin hole blocked off, plus a bump on one edge, so that it fits into a 39-pin connector on the mobo or HDD only one way. On the cable, the BLUE connector on one end goes to the mobo IDE port, the BLACK connector at the other end really should go to the Master device, and the GRAY one in the middle to the Slave device (if there is one). Use jumpers on each device to set as either Master or Slave, and remember that there MUST be a Master on the cable.
Exception: you can set the jumpers of BOTH devices on one cable (or the only device) to "CS" for Cable Select. Then the one connected to the END (Black) connector WILL be the Master.
IF you connect both a HDD and an optical unit to one cable / port, it is preferred that the HDD take the Master role.
Once you connect an IDE HDD to your machine you should check a few things in BIOS on your first boot-up. On many systems you do this by holding down the "Del" key when booting, but watch for screen messages that specify a different key to use to enter BIOS Setup. When in Setup you will have "tabs" across the top for major groups of functions, and many lines on each screen to check or alter settings. Usually the first screen shows you the main devices your machine has that have been detected, including all the disk drives. If your new IDE unit shows up here with reasonable-looking settings and a size that seems right, you're OK at the hardware level. If it does not, you should check further. In another menu you can configure the IDE and SATA ports. Make sure that the IDE port you are trying to use is Enabled. Once your BIOS can actually detect the existence of the IDE drive, check the screen where you specify the Boot Priority Sequence. Unless you plan to use this unit to boot from, the Sequence should NOT contain any reference to this IDE drive so that it never interferes with normal boot-up. If you changed anything, use the Save and Exit route to save settings and reboot with this configuration.
If your old IDE drive installed now in your machine cannot be detected properly in BIOS Setup, it has a hardware problem and will never be seen inside Windows (or whatever OS you are using). BUT if it is OK in BIOS, that does not guarantee it will work in Windows - there still can be software issues to fix. At least it can narrow down what you do to solve problems. Can't be seen in BIOS = hardware problem that must be fixed before anything further. Check cables (data and power supply), jumpers, try another IDE port, maybe your HDD itself is bad. CAN be seen properly in BIOS = all the hardware is OK, sort out software issues, Partitions, Formatting, etc.