How can I delete junk mail from Gmail

You could enable Mail's filter as well if you want. If you do so, I suggest you tell Mail to use Gmail's Spam folder as Mail's Junk folder. Do this by selecting the Gmail Spam folder, then go to the mail menu bar under the "Mailbox" menu "Use This Mailbox For" -> "Junk". This way, when you mark a mail as junk within Apple Mail, it will move to the Gmail spam folder, thus telling Gmail it is spam.

Alternatively, you could use Rules to kick out some of those mail messages you routinely get particularly if they always come from the same e-mail or have the similar subjects.

Gmail uses a concept called labels, rather than folders. When you delete a mail item, you are simply removing the inbox label, but it will still show in All Mail unless you delete it from there. I not a fan of the All Mail label, so I chose to hide the label from the Gmail account settings. To hide All Mail, log into Gmail on the web, access your mail settings, and go to the Label section. You can choose to hide any of your labels.

If your mark as junk is greyed out, then you don't have the Mail junk filters enabled (it's in your Mail prefs). I don't use Mail's junk filter because I let Gmail handle it. If you let Gmail handle it, then you will need to log into Gmail directly on the web, and mark such messages as Spam using the spam button. Gmail should learn your spam and prevent those from coming into your inbox (goes right to the spam folder).

Are you sure the inbox isn't at the top of your mailbox listing? You won't see it in your Gmail section:

You can mark a message as spam by right clicking on it, and saying mark as -> junk.