You didn't say what version of the MacOS you are using (and therefore what version of Apple Mail), but they're mostly similar.
1) Yes, to do any formatting your message needs to be "rich text"--the alternative is plain text, which is just what it sounds like: basic text, no formatting. When you are typing a message, at the bottom of the "Format" menu there should be an option that will say either "Make Plain Text" or "Make Rich Text"--if it says the former, it's already rich text and you should be able to add bold or underlined text in the message. If is says the latter, select that item and you will be able to format the text.
You can set the default for new messages in the Preferences, under the "Composing" tab.
2) Edit menu --> "Spelling And Grammar" --> Check Spelling --> While Typing. "While Typing" should have a check mark next to it, in which case Mail will underline misspelled words as you type (right-click or command-click on misspelled words to get suggestions of correct spelling). If it does not have a check mark, select it and it will from then on.
Or, if you prefer, you can select "Before Sending" to do just that. You can also select "Check Document Now" to bring up the spelling panel; command-semicolon (

will do the same thing.