What is the meaning of रहे?
"raha", "rehi" and "rehe" are words that you use to change a verb to the present continuous. Its changes something like "I eat." to "I am eating.".
Could this alternately be हम आज तैर नहीं रहे? I know in the simple present the auxiliary होना tends to get dropped
Re: Dropping auxiliary, yes.
Re: The change of word order that you made - That's OK, but it suggests a particular emphasis on "swimming", for example in contrast to something else that might have been discussed in the conversation.
Why it is not "Today we are not swimming"?