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First: this is all basic C++ and should have been asked on a more suitable forum (like Stack Overflow or similar). If you're new with C++ (and ROS), I would really recommend not trying to learn the two at the same time, as it's going to complicate things.

Having said that, see below for some comments on why the compiler is complaining.

In function ‘void diagnosticCallback(const DiagnosticArrayPtr&)’:
/home/default_ws/src/err_pack/src/errors_node.cpp:52:21: error: invalid use of ‘boost::detail::sp_array_access<diagnostic_msgs::DiagnosticArray_<std::allocator<void> > >::type {aka void}’
     if (diags_msg[0].level == diagnostic_msgs::DiagnosticStatus::ERROR)
                     ^

diag_msgs is of pointer type, so right now you're indexing against the raw pointer itself, not what it is pointing to (which is a diagnostic_msgs/DiagnosticArray).

If you want access to the array with messsages, I believe you should also be accessing the status field.

/home/default_ws/src/err_pack/src/errors_node.cpp:54:7: error: ‘errorcode’ was not declared in this scope
       errorcode = NodeToErrorCode[diags_msg[0].name];
       ^

There is no errorcode variable anywhere in the code you show, so this is the compiler telling you that.

/home/default_ws/src/err_pack/src/errors_node.cpp:54:19: error: ‘NodeToErrorCode’ was not declared in this scope
       errorcode = NodeToErrorCode[diags_msg[0].name];
                   ^

There is no ‘NodeToErrorCode’ variable anywhere in the code you show, so this is the compiler telling you that.

/home/default_ws/src/err_pack/src/errors_node.cpp:54:47: error: invalid use of ‘boost::detail::sp_array_access<diagnostic_msgs::DiagnosticArray_<std::allocator<void> > >::type {aka void}’
       errorcode = NodeToErrorCode[diags_msg[0].name];
                                               ^

This is the same problem as on line 54.

(I'm going to skip the other occurrences of this)

/home/default_ws/src/err_pack/src/errors_node.cpp:55:18: error: ‘requestStart’ was not declared in this scope
       requestStart();
                  ^

That function is defined later, so the compiler can't find it at this point.

Additionally requestStart(..) takes an errorLib::errorH& argument that you are not providing.

/home/default_ws/src/err_pack/src/errors_node.cpp:60:21: error: ‘requestNotStart’ was not declared in this scope
       requestNotStart();
                     ^

Same problem as on line 55.

/home/default_ws/src/err_pack/src/errors_node.cpp: In function ‘int requestStart(errorLib::errorH&)’:
/home/default_ws/src/err_pack/src/errors_node.cpp: In function ‘int main(int, char**)’:
/home/default_ws/src/err_pack/src/errors_node.cpp:93:53: error: no matching function for call to ‘errorLib::errorH::errorH(const char [21], int)’
   errorLib::errorH c("For starting/ending", 2);
                                                     ^

The ctor you've defined takes three args (EC value, string text, int sec), you're supplying it with two.