Opening the lid is fine. Keep as much sanitized as you can (I keep a squirt bottle filled with starsan solution for just this purpose). No need to stir, just open up and pull your sample.
I like my wine thief but have used a baster in the past. They both work.
You will want to calibrate your hydrometer anyway, so fill whatever tube you're going to use with distilled water and test your hydrometer. After you remove the hydrometer, mark the level with a sharpie or something so you know how full the jar needs to be. I use this jar from Northern Brewer:
because it is very easy to clean and it costs less than $5
Use your baster or wine thief to fill the jar to the mark and you won't have any spillage or extra waste.
Oh...and I just store my hydrometer in that tube under the sink. I haven't broke one yet.