Welcome back Mike,
I'm glad you did two things, (1) got the dish move, and (2) got someone else to do it.
There are a bunch of exceptions, but you will need to receive 2 satellite signals for basic programming for Dish - 119 and 110.
I believe the majority of local channels are on 110, but there are some on other satellites.
Most internationals can be found on 61.5, 148, 121, and it looks like 118.7.
High Definition programming can be found on 129 or 61.5, with a couple of programs on a couple of other birds.
When the tech set up your dishes and peaked them, I'm sure he gave you enough dishes for whatever satellites you were required to receive, which depends you what programming packages you ordered and what city or zip code you live in. I'm sure he checked the signal strength on each.
There is a very common dish called the Dish 500 that can receive two satellites, provided it has two LNBF's, or a Twin LNBF. I believe you said you have a second dish also. I will assume this other dish is another Dish 500 but with a single LNBF for one satellite, or maybe a Dish 300 or equivalent.
When you are watching a channel on TV, if you go to Menu, 6, 1, 1, it will give you the signal strength on a particular satellite and transponder. This is usually for the channel you were just watching. Not what the tech used to peak the system. Ideally, the tech would use a transponder that is not a spotbeam and/or would use a weaker transponder from each satellite he is peaking.
While in Menu 6-1-1, you can select the check switch button, and it probably will show you what satellites your receiver recognizes, that is, the satellites it found when the tech set everything up and did a check switch.
You will probably see 3 satellites listed, with more info. Such as:
119 110 61.5 or maybe
119 110 129
If you like, you can tell it to Test when in the check switch, and see what it does. It shouldn't hurt anything.
To answer your first question, yes, signal quality could affect the HD picture, but it would usually be in the form of pixalation, or no picture at all.
EDIT: When in Point Dish (Menu 6 1 1) and looking at the satellites, such as 110 and 119, or even 61.5 and 129, you will not get a signal on all transponders. So after you switch from 110 to 119, you may have to change transponders to find one with a signal. Also instead of going to Check Switch as I said above, you could switch from one sat to the next to see if you have a signal - but you need to wait about 8 - 10 seconds between each change to make sure the receiver has time to detect and report back to you, and try several transponders on each sat for best results.
If you think something is wrong with your HD Picture, maybe describe it and someone else will have some more info on that.
I think anything better than black & white on a 19" TV on a UHF channel 75 miles away is good!