Welcome! There may be more than one answer to some of your questions, but I will give you some possibilities to think about.

1. Begonias - could be cold damage. They like warm weather and we have had some cold nights. If that's the problem, they will recover and make new leaves.

2. Privet - prune it. It will make more branches if you do that. And it may need more sun.

3. Sky Pencil - it's supposed to be tall and leggy. It's a skinny plant. It likes more water than the sedum, so they are not good companions in a pot.

4. Althea - it will be just fine. Don't forget to water it. It needs water to make new leaves.

5. Roses - they don't climb or hang on to the wall by themselves. You have to attach them, but please don't use tape. Put screw eyes or nails into the wall and then use stretchy fasteners to attach the canes. You don't want to use string or wire or anything that will damage the canes.

6. Dog - if you can figure that one out, bottle it and make a million bucks!

2. "Chinese Varigated Privet - planted last summer. Very leggy and I'd like it to be more bushy and full. Any ideas?"

Yes, Chinese privet is one of the top 10 most problematic invasive species in GA. You should return it to the place of purachase for a refund citing that it's an invader of natural areas in our state and invasive species like this are creating $250 million annually in the US alone. As responsible gardeners we need to send a strong message that greedy growers distributing weeds is unacceptable. At the very least, just remove it and throw it away.

3." SkyPencil Holly (potted) - looking tall and leggy (not many leaves on the top). Its all green, just not very lush. Its in a pot with some dragons blood sedum. Any ideas how I can make it look fuller? Is it possible my pot is too small?"

The pot may be too small. It is a skinny plant as was mentioned but you could try doing some trimming to encourage growth in areas you want it more bushy.

4." Climbing Roses ..."
Needs something to cling to as was mentioned.

5." Suggestions for getting my dog to quit digging up everything I plant? ;-) "

Yes, I do. Supervise when your dog is in the yard and excercise their digging instincts on a regular basis in acceptable areas. Especially terriers. They were bred to hunt and dig for vermin. You may create a designated area like a sand box and bury chew toys known as Kongs in there. Stuff them with something yummy that smells good to dogs and encourage digging only in the sand box. They are rewarded with the treat when they find the kong and dig it up. No reward for digging in the wrong place. Backyard play is not a substitute for daily walks and mental stimulation and your dog will let you know how bored it is by the number of plants it digs up. Hopfully I got a million bucks coming my way but every time I dole out this advice I just get yelled at and accused of being rude.

Hey dandth, welcome.
The suggestion to use eye hooks in the wall for your roses is a good one.

About the privet - being new to gardening you couldn't possibly know that privet is one of the most invasive plants in Georgia. I can't believe someone is still selling Chinese privet. The solution is to rip and dig it out and make sure you get all the roots. If the seller won't take it back, destroy it. Every year the state parks in Georgia hold privet pulls to get it out of the wild. It's nasty stuff. So sorry you got suckered into buying it. Hope it won't discourage you from gardening to hear such bad news about your shrubs. Hope it won't discourage you from visiting this forum again and again.

I used nails to attached the roses to the wall..tough work - its a concrete block retaining wall.

I came across some very fine spider web attaching itself to the rose from the wall. I pulled it away. Do I need to treat it for insects now?