Which is better to use around the house foundation mulch or rocks? Should landscaping fabric be used for both?

Do not use rocks. They are hard on plants, give no nutrients to the plants and make it very hard to replant. Use mulch. I don't use landscape fabric with mulch.

If you do decide on rocks for any reason, then I would definitely use a heavy landscape fabric. If you ever want to remove or replace the rocks it will keep the soil and rocks separated.

We use both landscape fabric and cedar mulch around our foundation. It keeps unwanted plants from sprouting up, and keeps things neat and orderly behind our foundation evergreens also. But the landscape fabric is really unnecessary, except for reducing the errant plants from taking root or weeds coming up.