First, let me say Congratulations on your rescue! That's a wonderful thing you did.
Now, let's look at your dog. She is only 9 months old, which means she is a "teenager". She will be pushing the limits to see where her place is in her new "pack". That's you and your family! Being firm, consistent and patient at this time in her life will be very important. She is also part Lab! Labs are notorious "chow-hounds". They love to eat. (I've owned 4 so far!) So, even though this may be a tough time in her life, BOTH of these problems Can be solved, so here goes...
First, the tash picking. You'll need 3-4 empty soda cans. Place 3-4 pennies inside each can and tape up the opening. You can try something called "sound aversion". For now, only use these to correct the trash picking, until she has stopped that behavior. Clean out the trash can, put a new bag in. Now put a nice smelly morsle in the trash and walk away. Watch your dog. The moment she goes toward the trash, throw a soda can behind her (never AT her), she'll jump and (hopefully) turn to see what the sound was. At this point, give her the command "LEAVE it". If she turns away from the trash, praise her! What a good dog!! Let her relax, watch her, if she moves towards the trash again, repeat the steps. Keep it up until she no longer goes for the trash. Reward her for being so good! Repeat this process a few times, perhaps each day, if you still see the behavior. (putting the trash up out of the way when you aren't there will help keep her from making a mistake when you can't correct her) Keep your soda cans at the ready when your putting food into the trash. Soon, she won't want to go near that "noisy, scary" trash can.
Ok, on to HER food. There are a few things I would check here. First, try getting a good quality dog food. I recommend an all natural food. This doesn't mean she'll like it more, but it will be healthy for her. Check the "recommended" amounts stated on the bag, you can almost always cut that amount in half...I have found that their amounts are always too much! (adjust the amounts to suit her activity level and weight) Feed your pup twice a day. Once in the morning and once after YOUR dinner. Now, I have to say, I love giving my dogs table food. (no onions~they cause anemia~very bad) But I don't give it WHILE I'm eating. You could put it in with her dog food and she'd love you for it. But this gets done only after you are done eating. Also keep an eye on your dog food's expiration date and how long it takes you to go through a bag of food. Don't keep any bag longer than 4 weeks! Rancid dog food can cause cancer, so keep it fresh!
Ok, begging at the table. Let me just say, I don't believe in "tying a dog outside" for any reason, but here especially, it will never solve your problem. I would teach your dog a new command. "GO BED" Give your dog a small throw rug, dog bed, or towel. Name it "bed". Using treats and praise, teach your dog to lay down on this item and give the command "Go Bed". Once she knows this command you can use it at dinner time. Move the "bed" to a doorway or corner of the room where she can watch what's going on in the kitchen and dining room. Tell her "Go Bed", when she does, tell her she's a good girl (don't over do it and get her excited~the goal is to have her laying quietly on the bed), go on to making dinner etc. If she gets up say "AT, AT!" Give her the go bed command again, praise her when she lays down. Keep it up until she gives up trying to get up and stays on the bed. When you're done dinner, give her a "release" command like "OK!" or "your free". This gives her something to look forward to and tells her for sure that she is allowed to get up! Then, take a few healthy scraps (meat w/o too much fat, veggies, fruits) and mix up her food in her dish. Make her sit down before you give it to her. This shows her that you are still in control of what she eats. She will learn to wait patiently just outside of the room you are eating in or in the corner and that when she does, she gets a great dinner/breakfast. If food falls on the floor while you're eating and she goes to get it, give her the "leave it!" command that you've taught her with the trash. Tell her to "go bed" and when you are done dinner, give her the release command and she can "clean up" any droppings.
Well, I hope this helps! I'd love to hear how it goes, so feel free to email me! Good luck!!