Hi EggPharm! Welcome to LT! Regardless of what day of the week your contest ends on, you will get flooded with new designs during the last 24 hours of the competition. While you probably will get more entries on the weekend, we designers never sleep, so you will also get them on the weekdays.
Personally, I think you'll have more success with a short contest. You can always extend if you're not satisfied (though I don't know anything about whether or not there are additional costs based on the length of the competition). With a shorter contest, you'll probably get more quality designs early on, and you'll have fewer issues with last minute entries looking a lot like the highly ranked designs.
The key to a successful contest is ranking often and providing feedback to designers. The more feedback we get from you, the better your logo will be. Good luck!