Re: mehram/na-mehram
I dont know about the rest but @ your last sentence, yes, I have also heard that in Islam, adopted son can neither be given the name or share in inheritance. I mean he will not have a share automatically as good as own children. But that is something else.
The OP asks about the issue of mehram and na mehram in adoptation. YES it exists since obviously a boy 10 years plus is na mehram so obviously you cannot make believe that you have adopted a boy and he will be your son for like forever and you and he both can treat each other like real mother and son, NO YOU CANT. I think its in Surah Noor where explicit details of mehram and na mehram relations have been defined. Obviously the same goes if you adopt a girl. Your husband will be na mehram for her once she comes of the age (I am not sure whats the exact coming of age is in shareeah).
The point being, the basic idea of adoptation is not much encouraged in Islam. Maybe for the fitna it could create. Like this issue of mehram and na mehram. This issue wont exist if you:
Either are living alone and you adopt same sex baby or a nephew/niece already.
Or as a couple you adopt a baby which happens to be a mehram already. Like you can adopt your husband’s niece or your nephew.
If you want to adopt for sawab only, you can always support a family or a kid’s education financially.