When connecting to Oracle DB I get External: SIGSEGV error

The fact that the change I suggested in my comment, namely

if (OracleMng <>  Nil) and (OracleMng.Zquery1 <> Nil) then
  OracleMng.ZQuery1.SQL.Clear

evidently stopped you getting the SIGSEGV error suggests that your DataModule and form are being created in the wrong order, i.e. form first. Check this out by going to Project | View Source in the IDE. If you see something like

program MyProgram;

[...]

begin
  Application.Initialize;
  Application.MainFormOnTaskbar := True;
  Application.CreateForm(TForm1, Form1);
  Application.CreateForm(TDataModule1, DataModule1);
  Application.Run;
end.

they are in the wrong order, so swap the two CreateForm lines

  Application.CreateForm(TDataModule1, DataModule1);
  Application.CreateForm(TForm1, Form1);

With that change, you should no longer need the

if (OracleMng <>  Nil) and (OracleMng.Zquery1 <> Nil) then`

Next thing: You seem to be confused about when to use

ZQuery1.ExecSQL

and

ZQuery1.Open

Open is intended for when the SQL statement you are using produces a result set, that is a collection of records which can be viewed in a TDBGrid. The most usual way to do this is to use a SELECT statement as in

ZQuery1.SQL.Text := 'select * from MyTable';
ZQuery1.Open;

ExecQuery is intended for use where your SQL statement performs some operation on the database which does not involve SELECTing records. The most common SQL statements which need ExecSQL are

UPDATE
INSERT
DELETE

though there are others, for example statements which execute stored procedures on the SQL Server (note that some stored procedures return result sets and so need Open, rather than ExecSQL).

Note that ExecSQL will clear out any records which are in the dataset (ZQuery1) so after you need to do Open again using a suitable SQL statement

var
  S : String;
begin
  S := 'update MyTable set number = number +1 where id = 5';
  ZQuery.SQL.Text := S;
  ZQuery1.ExecSQL;    // no records shown in DBGrid1 from here

  S := 'select * from MyTable';
  ZQuery.SQL.Text := S;
  ZQuery1.Open;    // records shown in DBGrid1 again
end;

Note that I do

  S := 'select * from MyTable';
  ZQuery.SQL.Text := S;

instead of

  ZQuery1.SQL.Clear;
  ZQuery1.SQL.Add('select * from myTable');

The reason for this is that it's much easier to see the whole SQL statement in the debugger by inspecting the variable S than inspecting the ZQuery1.SQL.Text property and much easier to see any syntax errors.

You should always Close a dataset that you've Opened once you have finished working with it as it ensures what the data on disk is up to date. if the last SQL operation was ExecSQL, you don't need to close the dataset.

If you set the query's Text property the way I do, with ZQuery1.SQL.Text, you don't need to uses Clear. In any case, it is only equivalent to doing ZQuery1.SQL.Text := '' and it does not affect the state of the dataset - it only does anything when you call ExecSQL or Open.