Above statement is correct. The quarantine period between new and old birds should last about a month at least unless a wellness exam and any diagnostics are completed to rule out possible health issues. This is to avoid your old bird getting sick by the new birds possible air borne contaminants, if present at all. So make sure you separate them when you can.
As for your questions...
1. New birds will naturally feel very stressed as you know when they enter a new environment. He will eventually eat so donāt worry about that for now. During the day when your birds cage is uncovered, keep the towel away and instead, step out of the room if you can because he may be stressed from a lot of the interaction happening if he is untamed. Even looking at the bird or being in the same room can make an untamed bird very difficult to handle, but there are solutions. Try to...
A) Step out of the room when possible until he is relaxed.
B) Stay in the room but try to not make sudden movements and do not look at the bird until he is calm.
C) If he is still not relaxed, do something else for awhile and let him relax on his own.
It also helps when the birds cage is in sight from the entrance of the room so that your bird does not get any quick surprises when you do come in. And when you walk by the room he will see you come and go. Just let time take the lead and youāll be good to go.
2. For your other question, just make sure they are in separate rooms for quarantine as stated above, and you shouldnāt have any problems there.
After quarantine: Here are some tips
1. Let your untamed bird come out of his cage on his own accord by simply opening the cage door for him while you are in the room supervising.
2. Bring your tamed bird closer to you or let him perch on your hand and let the other bird see that. Birds are smart creatures. The untamed bird will see that and could think you are not as much of a threat as he assumed you were.
3. Use treats as rewards to positively reinforce good behavior from your untamed bird. This will just strengthen your trust building efforts and let your bird see that you really mean no harm.
4. Have a routine with both of your birds together so that your new bird will know what to expect and will calm his stress by a little bit knowing that the ānext part of the routineā is coming up. (If you are consistent)
Note: Do not expect your birds to get along. The goal for us is for them to tolerate each other and each otherās presence, if they continuously fight and hate each other, thatās a problem. But just keep the routine and keep supervising for extra caution.
Sorry that this post is long and apologies for referring to your birds as ātamedā and āuntamedā lol I forgot if you mentioned a name at all. And yeah thatās all.
hope u having a good day, so i brought an untamed cockatiel to Pika (my tamed cockatiel), i have them in the same room separate cages, my tamed cockatiel is normal as usual but the untamed one is still stressed from yesterday. I gave him some treats but no luck he is still very stressed. Whenever i come close to him hes panicking, as soon as i cover the cage with a towel he stops panicking. Should i keep the towel until hes more relaxed? Im scared that he might not eat because of stress. Any recommendations about what to do? Can i have my tamed cockatiel outside the cage while the other is in the cage? 
