Common name =- cock's comb.

Looks like a florist's gift plant that is only meant to tolerate indoor conditions for a limited time and then be discarded.

I agre with rina. My grandma called it a cockscomb. It's an annual and not usually a good houseplant choice.

Is it a true annual or just sold as such in climates that have cold winters?

They're true annuals afaik.

In that case, it can't stay alive indefinitely in any location or in any conditions.

Came across sources indicating Celosia to be a tender perennial (zones 10-12). With such a limited perennial zone, probably the reason it is thought to be an annual. Interestingly enough -- to me at any rate -- is reading that it is an edible plant -- leaves, young stems, and seeds.

Yes, I did too. I find the phrasing of some of them frustrating though.
"USDA Zones — 3 – 11, perennial in zones 9 and 10"
Taking a perennial to where it can't survive perennially does not make it an annual, and listing this plant for z's 3-11 is bizarre. The zones exist to indicate where a plant is hardy. This plants' zones are 10+, or could just be called tropical. Oh brother, I just realized the one with that phrasing (3rd link below) seems to be connected to this site in some way.

http://edis.ifas.ufl.edu/fp114
http://aggie-horticulture.tamu.edu//newsletters/hortupdate/hortupdate_archives/2006/apr06/Celosia.html
http://balconygardenweb.com/how-to-grow-celosia-growing-and-planting-guide/
http://homeguides.sfgate.com/celosia-plant-care-information-35143.html

?

Here is another source of good information. Celosia argentea var. cristata (Cristata Group) - Plant Finder - Celosia argentea var. cristata Common Name: cockscomb Celosia is a tender perennial that is winter hardy to USDA Zones 10-11.

So now that is has been established that it is worthy of your efforts to keep as a house plant, you can do your best to meet its' needs throughout the year. I don't feel qualified to advise, not a plant I've played with before, but they are a popular plant during summer in most places, I guess, I've seen them for sale my whole life. Assuming you're at a higher, temperate latitude, know that shorter days and weaker rays of winter at higher latitudes can mean that a plant that thrives without any direct sun during summer could need some over winter.

Give only grown these outside and always in full sun.

This is conjecture only, but, if your indoor humidity is very low, you might have issues with spider mites and leaf drop or leaf margins browning out. Seems like many plants with broad, thin leaves have such issues when grown indoors.