When I slow down the tempo of a song that has samples in it (samples cut from other songs not loops) the samples start to clip. Shouldn't changing the tempo of a sample just change the pitch? Is Ableton automatically preventing this, which is causing them to clip? How can I prevent this?
There's several different warp mode in Live. Have a look at the cli view panel. If the clip is unwarped, slowing down the tempo will pitch the clip down and make it longer, and vice versa, like in the analog world. If the clip is warped Live will stretch it but keep the original pitch. Those "clipping" you mention are because your clip are warped with the "beat" mode.
If you have a sample loaded in Live, it generally is warped meaning that Live will generally attempt to preserve the pitch of it no matter how you stretch it in time. This is of course impossible to do perfectly, but it uses some very clever algorithms to attempt it by making educated guesses about the data. The different algorithms are referred to internally as "warp modes."
However, these guesses are an attempt to recreate data that isn't there - and one possible outcome is that it guesses that the waveform may be higher in parts than in your original sample! So a "warped" sample can peak slightly higher than the original sample would.
This is true for most warp modes with the notable exception of "re-pitch" which does not attempt to guess anything - it simply moves through the original sample at different speeds.
In general it's a good idea to leave yourself some headroom when working with any recording - you can always turn up your speakers if it's too quiet.