The dialog box for entering the URL for a hyperlink (shortcut Ctrl + L) should be available to you. It's just a window with one text entry box where you can enter the URL. The box is initially filled with the text "http://", but that should be replaced if you type or paste something into the text entry box.
I don't know about the other dialog boxes. I think you could completely skip the dialog box to upload an image or document if you can use drag 'n drop to do it. As for the smiley emoticons, perhaps just stick to using the simple, basic text based ones such as
:( for now?
Out of curiosity, what does your screen reader do when it bumps up against an emoji?
In addition to Markdown, the Discourse editor will also accept some "safe" (?) subset of HTML and a "basic subset" of BBCode. So I think you should be able to use pretty much the same approach for formatting which worked for you on the old site should you wish to.
But learning Markdown might also be helpful to you. If you understand how Markdown works then you would recognize the meaning of the Markdown markup tags which Discourse wraps around text to format it.
Some very simple Markdown markup examples.
If I want to display the word "rutabaga" in italics (rutabaga) then in your editing window Discourse will enclose the word with an asterisk on each side like this:
To get boldface rutabaga Discourse surrounds the word in the editor with two asterisks like this:
So ... how to get both boldfaced and italicised? Yepper! Three asterisks. So
***rutabaga*** in your Discourse editor is formatted as rutabaga when you post the message.
By the way, if you ever want to have Discourse ignore the "special meaning" of a character in your text, you can simply preface it with an "\". \ is the escape character (if I recall correctly).