I have a parameter for a date like 3 months, 6 months etc. but I don't want the current month to be selected.

On the filter pane, you can select the exclude current date. This can be done by checking the exclude checkbox and checking another checkbox showing you at the bottom written Filter to the latest date when the worksheet has been opened.

In parameter, whatever I enter for a fixed list, shows in the parameter. Is there a way to keep some other name?

Yes, this can be done with the help of the display name in the parameter. when you create a parameter, you get two options-

You want some other name apart from values you can put those names in the display as a section. Now the user will be able to see the display as name and not the value.