If you bottle a mead with any residual sugar content, it is wise to ensure that there are no more viable yeast in the bottles. This can be done one of several ways. First of all, make sure that all active fermentation has stopped. You may have dormant yeast in there, but if you have actively fermenting yeast (such that the SG is still dropping over time), then most methods to inhibit fermentation are prone to fail. Make sure that things are dormant.
The most common method to stabilize mead or wine, is to add some sulfite and some potassium sorbate shortly before bottling. The sulfites will generally prevent most dormant yeast cells from ever waking back up, and will also inhibit infection organisms and wild yeasts from taking hold. The sorbate will prevent any active cells that happen to get into your bottles from ever reproducing, so even though a few stragglers may still ferment some sugar, they usually don't amount to much, and then they quickly expire from the combined stress of ethanol and sulfite. It is important to use sulfites with the sorbate, since sorbate alone can cause other problems. You can search on the terms sorbate and geraniums to find out exactly what. ;-)
Alternately, if you have access to a wine filter and some sub-micron filter pads you can remove all the yeast from your mead. This is a rather expensive option for home winemakers, and in order to be absolutely sure that no yeast have found their way around the filter it is still a good idea to add sulfite and sorbate, so I would only filter if you have the equipment to do so.
Finally, pasteurisation is an option, but since that generally changes the flavour profile of the mead, I don't recommend it.
And yes, it is always a good idea (which means that I always do it!!) to rinse your bottles and corks with a sanitising solution. I use the no-rinse sanitiser, Star-San, since I don't want to compromise the state of my bottles by having to rinse them after treating them. However in years past I successfully used chlorine bleach, followed by a thorough rinse of plain water, with no ill effects. The Star-San is simply easier to use!