Rename Schema Removes Unique Constraints?

I decided last night to rename my 'public' schema (Not sure of that's
a good / bad idea) since I'm still learning about how schema's work on
PostgreSQL. My question is:

1. If I have a constraint (specifically 'unique') on a specific table,
when I rename the public schema, does that impact my ''unique', 'not
null', and 'primary key' constraints?

2. When you install a new PostgreSQL server at home or work, do you
guys generally leave the default 'public' schema or do you at some
point rename or use new schema for reasons I'd like to understand. I
don't understand how schema's really fit into the picture but I'm
interested in knowing what others do with their default schema.

Thanks for the help!