Cody,
"Or is the sealing just to keep water out of the thread-holes?"
The thread/needle holes will let rain water from those "small thunderstorms" through your tarp and onto you.
"Is it a problem if I use it in the rain once before sealing?" "Will that damage anything?"
Even one steady drip during the night could keep you awake and aware of just how much more comfortable and dry you would have been.;-)
If your ridgeline has a flat felled seam, which is four layers of silnylon sewn down flat, the water could infiltrate and enter this seam and get between the layers of material. It could take a very long time to dry out. It could also lead to staining of the fabric due to mildew.:-(
"I thought about keeping the bugs and little critters out by using a Bivy, just haven't gotten there yet…"
If there are flying insects in the area consider a bug head net. It may help preserve your sanity.;-)
If you sleep with the head net over a ball cap the bill will keep the net off of your face while you're sleeping.
"…I'll be using a tyvek like sheet as a groundsheet, so shouldn't get any ground moisture, but I'm a little worried about water running over the ground into my bag… Does this actually happen?"
It happens! Choosing your site carefully is very important. Finding a relatively flat and somewhat raised site is ideal but often not the easiest goal to accomplish. You could guy out and up the corners of of your ground sheet to the corners of your tarp. This would raise the edges slightly and possibly keep the water running underneath.
Be sure and keep the edges of your ground sheet 4" to 6" inside of the edges of your tarp. Otherwise rain water will collect on top of your ground sheet and under you!
You can also raise the edges of your ground sheet using available materials such as forest duff, leaves etc, to create a raised "berm" that will divert the water around and/or under your ground sheet. If you use this method weight down the corners/edges on the outside of your "berm" with some rocks.
The seam sealing should only take an hour or so. Let it dry for 24 hours. If it still feels a little tacky dust it lightly with some unscented talcum powder.
Enjoy your new tarp and your weekend adventure.
Party On,
Newton