Which properties should I look for or avoid in rice when I'm planning to fry it (the day after cooking)?
It depends how you like your rice. Personally I like my fried rice to have a few sticky lumps through it to add texture. Others may like it completely 'separated'. It's really up to you. I usually use a combination of long grain and Basmati.
Another point is to make sure your rice is safe - it needs to be cooled relatively quickly, and kept cool, to prevent the growth of Bacillus cereus, which can cause serious (no pun intended) food poisoning.
As it happens, fried rice is much better when cooked with cold rice and very hot oil anyway, so it's a good idea to keep it cool in any case.
Technically you could probably use whatever rice you have there. But if you're looking to make a typical Chinese style Fried rice, traditionally a long grain white rice is used. The long grain rice has the lowest amount of amylopectin which is what makes some rice "sticky". Typically when fried rice is made, you want individual separable grains of rice. So I'd go with the longest grain rice you have.
The most important property is that it's cold. Really.
Cold, slightly dry rice of any variety will do, except for highly glutinous rice (like mochi-gome). Mochi-gome, while used for mixed rices like sekihan (rice and red beans), is generally steamed together with other ingredients, not fried, because it'll just be uncooperative.
Japan, China, Korea, and Thailand all have some form of fried rice, and the variety of rice is generally the same as whatever the most common variety is in that country. Japan and Korea usually use a short-grain japonica variety which tends to clump a bit, and China and Thailand mostly use medium grain which generally doesn't. Generally, because the rice dries substantially during cooling, the clumping is greatly reduced when you toss that pre-cooled rice around in your frying pan/wok/cooking vessel with a bit of oil.