Where should one put indexes in a time dimension table?
You likely won't run into the issues of write problems, as I assume this would be something created once (or once per year), and then not touched.

But using an index will likely be a hinderance if you're searching by week ...  The problem is, if the index is used, it might scan that first, and then grab each record out of the table individually, which when you're pulling out more than about 5-20% of the records, it's typically faster to do a full table scan, and then drop the records you don't care about.

I don't know of any major RDBMSes that don't optimize for this when it's well-distributed data.  If it's not well distributed (eg, one of the values in a column occurs 95% of the time, but there's also other possible values), you may have to compute histograms on the table and not use a placeholder for the value when searching, so that the query optimizer has the value being searched for when generating the execution plan.

I'd likely not index day of week.  I'd check my database's documentation to see what their tradeoff is for indexed reads vs. full table scans to see if I'd index the day of the month or month of the year.  I'd likely index DOY/day of year if present (which sounds like it's your unique index, anyway)