Your CD burning software must be capable of converting them to 44100 Hz on the fly.
It is still best to Export as 44100Hz if you know that you are going to burn them to CD as this means that your CD burning software does not have to do it (for reliability it’s best to keep the demands on your computer reasonably low while you are burning CD’s).
Each track in an Audacity project can have it’s own sample rate and bit depth.
In Audacity 1.2.x you need to take care when copying audio from one track to another that both tracks have the same sample rate. If you move some audio from one track to a track which has a different sample rate, then it will play at the wrong speed and pitch. This is not so much of an issue in Audacity 1.3.x as Audacity can now convert the audio to match the sample rate of the track into which it is being pasted.
Higher sample rates allow higher frequencies to be stored. The absolute theoretical maximum frequency that can be represented in digital audio is half of the sample rate. In practice it is a little less. 44100Hz is used on CDs and allows them a maximum frequency response of up to 20kHz. With 48kHz sample rates, the frequency response goes a bit higher, and the response above 17kHz is more accurate than when using a 44.1 kHz sample rate (that is why it is often used in professional audio).
For high quality recordings you should not go below 44100Hz as your sample rate. As the sample rate goes lower, the sound will become dull and loose its sparkle.
When it comes to Exporting, it is the project rate that is set at the time of exporting that is important.
In Audacity you can have the best of both worlds - you can record and edit at 32 bit 48kHz for very high quality, then Export at 16 bit 44100Hz for compatibility. This is common practice in studios.
Note that above 48kHz there is insignificant improvement in sound quality - it just makes the files bigger.
Recording at 32 bit (the bit depth setting) offers slightly better quality than 16 bit, but all the files are twice the size of 16 bit files. The difference is unlikely to be noticeable unless you are doing a great deal of processing.
The best choices for high quality recording are either 16 or 32 bit and 44.1kHz or 48 kHz.
For compatibility, the best choice for exporting is 16 bit 44.1 kHz.
If you don’t want to have to remember to change the project rate, just do everything in 16 bit 44100Hz.
Regarding the CD problem:
My suspicions are now lying firmly with the CD drive in your computer.
If the CD burns properly on another computer, then try as an experiment copying a known good CD on your computer. If this causes the same problem, then you probably need a new CD drive. Fortunately CD/DVD writers are pretty cheap these days. Unfortunately it is probably too late for Santa to bring you one.