When you say "baby" how old will the chin be? If it's under 6 months or at least 400g then that cage will not work at all. The bar spacing for kits under about 4-6 months (about 400g) is 1/2" and 1" bar spacing for adults (over 400g), kits under about 400g can literally squeeze right through 1" bars, or worse, get stuck trying requiring you to cut them out. If the spacing is 1" you will need to wrap the whole cage in hardware cloth to prevent escaping. As for how high, ideally you want a low cage (no more the a couple feet tall) for kits until they are at least 6 months as well. No ledges higher then about 6" and make sure they are all 2" or less apart up/down and left/right to avoid a bad fall. A fall of more then about 6" can seriously hurt a kit if they land wrong, that also means you want to make sure they can't climb the cage very high either, kits will climb the cage like little monkeys, and some like to climb up as high as they can then just let go and fall
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Honestly though, that might work ok as a temporary cage for an adult chinchilla, assuming they don't chew the plastic, but over all that is not a good cage for chins. Even if you replace the plastic shelves, people in another chinchilla group I'm in have had chins chew right through the door as well as chew through the plastic connectors holding the cage together causing the cage to become unstable and risk collapsing, requiring a quick cage change out. I don't know if you could maybe make a replacement door out of metal, as well as somehow change out the plastic connectors for metal ones or cover them so the chin can't access them. As mentioned, how quickly or even if your specific chin will chew the plastic varies from chin to chin. Some decide to just chew the first day and you wake up to holes through the plastic, others can go weeks, months, or even in rare cases, their whole life without ever touching it. The biggest issue though is if they chew off and swallow a big enough piece it can cause them to choke or get stuck in the gut, requiring emergency surgery, if it doesn't kill the chin first, and it doesn't take a determined chin long to chew a large enough piece off to cause a problem. So if you are lucky yours wont chew the plastic before you can get a better cage, but be prepared to need to remove all plastic from the cage at anytime, which includes getting the chin in a new cage if they chew the connectors.
To answer your question, tile should be fine, as said unless you keep the room very cold, like below 60F, it's a good idea to get a thermometer for the room so you know what the actual temp is. As said with tile though, if the chin is not potty trained to pee in a pan, you do run the risk of it getting covered in pee from zooming through pee puddles. You might beable to put tiles down then some shavings over top though to help absorb any pee.