How can I allow anonymous visitors to submit content?
Create another content type, say "Event Suggestion", which contains only those fields you want anonymous users to submit suggestions for. Do not include the fields you consider as not relevant (and which you want to 'hide' for such anonymous users).

When creating such extra content type, for its fields just use the same fields as for the content type that you already have, say "Event". So the fields for this new content type (Event Suggestion) will be a subset of the fields that you already have for the content type already being used by your "small group of users" (= Event).

Related to permissions: allow anonymous users to create an Event suggestion, but do not allow Event to be created by anonymous users. 

For spam reduction/prevention, use modules such as Honeypot, Mollom, Captcha, reCaptcha, etc.

Use the Rules module to finetune your workflow related to nodes of type Event suggestions being created (saved). To just mention a few options (the list is virtually unlimited):


Unpublish such nodes (so that spam submissions that are not caught by spam prevention modules, never get shown without reviewing them first).
Send an eMail msg to your small group of users who manage your site (something like "a new Event suggestion is waiting your approval".
Set some kind of "confirmation message" when an Event suggestion is beiing submitted (like "Thank you for your Event suggestion, it will be reviewed soon ...").


When your small group of users who manage your site then review such Event suggestion, they should use the Node Convert module to convert an Event Suggestion into an Event. Using the Rules module you could then automatically "publish" such event also. After such node got converted, your small group of users who manage your site could then complete what you described as "add more details later on".

Other modules like Flag and/or Content Access might further enhance the various features / functionality you might be looking for. E.g. as an alternative to working wirth "unpublished" content, you could work with content that is published, but can only be accessed by authorized users.

Bottom line: Rules and Node Convert could get you pretty close, possibly further enhanced with Flag and Content Access.

Note: Adding the View unpublished module seems like an interesting improvement also, i.e. to grant access for specific user roles (such as editors or reviewers) to view unpublished nodes of a specific type.