As far as coming out of the crate.
Walk up to the crate. Tell her down. Wait motionless until she complies. When she does, move your hand to the latch on the crate door and touch it. Being a puppy, she will get up. Say nothing, but immediately remove your hand and look at her expectantly. She will wiggle for a few minutes, then maybe try and get you to let her out (cry, hop, etc). then... (yes it really is true) .. she will lay back down. Immediately put your hand back on the latch. If she remains down this time, let her out (but if she gets up at any time, remove hand again). Sometimes with my guy I had to undo the latch but hold the door closed with my hand because he would try and push out. Sometimes also, I had to open the door a couple inches then slam it shut because he got up. Now he knows he has to wait until I say "OK" before coming out of his crate or else I will slam the door on him!! (not on his nose. If his nose comes out I push it back in first, so I don't hurt him).
He has learned: stay down = get out faster. I never say anything to him or punish him, but if he isn't calm, he doesn't come out plain and simple.
This even works with shelter dogs the first day or 2 I have them. They catch on really quickly!
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Now as far as begging from the table. This is behaviour that should be corrected. I had a pushy pup too. (He is more attention motivated than food motivated, though). What worked is making sure he had a toy to play with, at first, something like a stuffed Kong that was REALLY good. I would be at the table and puppy on his rug with his bone. I always slowly, quietly and calmly said "yeeeeessss... gooood puppppyy... gooood chewing.... (etc)", then ignore him for a minute while I carry on my work, then prais again etc. If he bugged me, Stern face, "no", lead him by collar to his rug. Now (15 months old) if I am busy he will go find his own toy somewhere in the house, and I'll look up and he will be laying about 8 feet from me, chewing his toy like an angel! (I always praise calmly when I see this!) Make sure she has something to do other than bug you - She is mentally the age of a four year old child, so if her brain isn't occupied or sleeping, she is finding something to do. Waiting patiently is not in the vocabulary just yet!
You have the right idea - praise when she IS doing the right thing and if you can, before she starts "bad" behaviour.
I am also an "adult dog" person and swear I'll never have another puppy (even though they're cute!) - that age was the worst! Around 10 months old he became more tolerable - but then I work outside the home and didn't have as much time to spend with him as you do. I'm sure you'll see good progress soon.
If you want to get any Leerburg DVDs, the "Your Puppy 8 weeks to 8 months" and the marker training DVD are good choices.