Is it possible to open NERDTree in every tab with pressing t or T in NERDTree, if yes, How?
This is probably not the best way, but if you edit plugin/NERDTree.vim and change this:
exec "nnoremap ". g:NERDTreeMapOpenInTab ." :call openInNewTab(0)"
to this:
exec "nnoremap ". g:NERDTreeMapOpenInTab ." :call openInNewTab(0):NERDTree"
it will alter the binding of 't' in the NERDTree view to first open the file and then open NERDTree. Note, that the NERDTree views will not keep in sync.