No. Normal eSIM use never requires it, many handsets do not display it at all, and where it does appear the menu path differs by manufacturer and OS version. If you need to contact support, the useful references are your order number and the email you purchased with.

No. The IMSI is how network systems recognise your subscription; a phone number is the address other people use to reach you. They can be linked in an operator's database but they are separable, and plenty of data-only plans have the first without the second.

Because the name shown is the local network currently carrying your signal, while your account still sits with the operator that issued the profile. The local network reads the operator code from your identity and checks back with the issuer. That is roaming working as designed, not a fault.

Not until the seller confirms that profile can be reinstalled, because many download only once. Restart the handset, cycle aeroplane mode, try selecting a network manually, check that data roaming is on, and confirm mobile data points at the right line. If it still fails, contact support.

Because the operator that issued your profile already holds roaming agreements with networks in those countries. On arrival the local network serves you on that operator's behalf, so you do not swap profiles at a border. The experience can still differ country by country, depending on what agreements exist there.

No. Adding or removing an eSIM profile does not touch your photos, contacts or apps. What changes is the subscriber identity inside that profile. The one caution is that deleting a profile with allowance left usually forfeits the remaining data, and that cannot be undone.