TLDR: CDs and DVDs can be safely reused or recycled as long as you don't heat them. Specialized recycling companies can recycle them for you, but this may cost you money.
CDs and DVDs are mainly plastic (polycarbonate), with a thin reflective layer (usually aluminum), and a protective acrylic lacquer. CD cases are usually made from polystyrene and DVD cases are polypropylene. I've never heard of CDs or DVDs containing cyanide.
All these materials can be safely cut, but as you mentioned in your question you have to watch out for sharp edges and splinters. Some people heat plastics so they can reshape them (e.g. see this Youtube video), but I can't recommend this because harmful fumes may be released when plastics are heated (see also this question about melting and molding plastics)
Reusing CDs and DVDs is more environmentally friendly than recycling as no energy is needed to convert the materials to usable resources, but it requires some effort on your part. You can:
- sell you old music CDs or video DVDs, or give them away on websites like The Freecycle Network
- use them to create art (mosaics, ornaments)
- turn them into a clock, or light
- reuse them as coasters.
More reuse suggestions can be found here 10 Reuse Ideas for CDs and DVDs or in this Youtube video. I'm sure that if you search the internet you can find a whole lot more.
There are specialized companies that recycle CDs, DVDs, and sometimes also videotapes or computer hard drives, but often you have to pay them and not all companies accept materials from households.
If you live in the US, there are a number of recycling centers where you can drop CDs and DVDs. You can check the recycling locator on the Earth 911 website for addresses. Some companies allow you to send CDs and DVDs and you only pay for shipping, e.g. the CD Recycling Center of America in Salem, NH and Back thru the future CD/DVD Recycling in Franklin, NJ.
Recycling centers usually ask you to separate sleeves, casings, paper inserts and discs, but regulations may vary per recycling center.
The Recycling Center of America made this Youtube video showing how they want you to collect and send everything, and how they process it.
I know there are also specialized companies with drop-off points or mailing possibilities in many European countries. If you live elsewhere I suggest you contact your local recycling center and ask if they have a specific recycling scheme for CDs and DVDs (other than incineration). Or you can search the Internet for specialized recycling companies or drop-off boxes in your area.