First thing is if you are using dry yeast, you should really rehydrate it correctly according to the yeast manufacturers instructions. A lot of people will just sprinkle it on but the yeast is unable to regulate what passes through its cell wall for the split second or so after it hits the wort when dry and you can lose a lot of the yeast that way. That being said, if you pitch a big pack into an average gravity wort, you still would have enough yeast to do the job most times. There are a lot of differing opinions on this topic so I would try several batches both ways and use what works for you in your brewery.
Don't sweat opening it up and observing and trying it. At this stage you should be more concerned about understanding what is going on and using your observations so that you build a frame of reference for future brewing. You won't be hurting the beer by opening it up and drawing off a sample. Just make sure to sanitize everything that comes in contact with the beer in the fermentor, and try to minimize exposure time for the open lid.
It doesn't take a few weeks for the gravity to get to the FG. You should be at FG usually within 3-5 days if you pitched the correct amount of yeast, added oxygen and had temp control. At 62 things will be a little slower than 67-68 but you should still be close. Then let it set for a few more days to a week longer to let the yeast clean up any extra by-products and drop clear. Use your hydrometer and taste to guide you from that point. The key is to build up a frame of reference of what a beer should taste like at any given point to be able to understand when is the right time to do the next step.
There are a lot of different ways to add oxygen. I use pure O2 with a stone, some shake, some use aquarium pumps. Check all of them out and decide what works best for you in your brewery with your budget.