Gaiwans definitely can be unglazed. Case in point:
Of course being unglazed means you either dedicate it to one type of tea or clean it every few months with baking soda. This one I use for shengs.
If you're unsure what to buy, a white porcelain gaiwan is always a good choice. They're as neutral as it gets, come in a variety of sizes (usually in the 90-160 ml range, but I've seen smaller ones), and they're inexpensive so even if you break them it's not a big loss.
Pouring from a gaiwan can be tricky, but if you hold it properly by the rim and don't put too much water you won't feel the heat. Well alright, you might still feel it, but it won't burn your fingers.
Also important to remember that gaiwans are not strictly reserved for gong fu brewing. You can have more relaxed sessions with a gaiwan too, especially if it's a large one. I find it's still better than a single, long brewing in a Western pot or in a French press.The only teas I do gong fu style is sheng and shou and maybe heicha. I'm thinking about getting some type of french press for work/travel and keeping the clay gaiwains and teapots at home.