How to find the all positions of first element and get the elements at once from the nested list?

Thread[Extract[expr, #] -> #] & Position[expr, _Symbol, Heads -> False]

{plus -> {1}, minus -> {2, 1}, times -> {3, 1}}

expr[[Position[expr, {minus, __}][[1, 1]]]]

{minus, 2, 3}

Also

First@Cases[{minus, __}]@expr

{minus, 2, 3}

For the second question:

expr[[Sequence @@ Most @@ Position[expr, minus]]]

{minus, 2, 3}