I’ll try my best to answer what I know…
I learned how to ride on a Morgan and competed for several years on the circuit, ending in 2006. I have owned Morgans for over 10yrs. The only flightly/“non-sensible” Morgans I have met have been bred as show horses. I will add to this around a month ago the AMHA released an article requesting judges to return to the original standard and stop pinning wavy-leg, shaky-tail horses. As you can tell I’m not really a fan of the current winning show Morgans.
Anyways…back to your questions. I’ve seen several excellent lower level dressage Morgans.
I find Morgans are more “hot” in that they can be sensitive. Generally they will not take to drilling on the same movement over and over and over again. They will begin to anticipate and get “hot”. I’ve also ridden many a dull sided lesson Morgan, but they still halted when the trainer said walk and we were cantering!
Lippitt Morgans tend to be very classically bred but with a heavy shoulder and neck. They are generally the pony sized up to 14.2. They have created their own club and tend to think they are the “true” Morgans.
The Government bred line is a tricky bit. My experience with them has been through the University of Vermont Morgans (not actually related to the University sadly). They are in the 14.2-15.2h range usually and in my experience tend to be a bit lighter build than the lippett, or brunk/western working. Great minds but more geared towards driving and saddle horses. However, many sport horse geared Morgan’s have UVM Government lines, such as Taproot Command Pilot. If I remember correctly FEI rider/trainer Madeleine Austin worked with him and thought him a lovely horse.
The brunk/working western Morgans are actually way back in the heritage of QH’s. I feel they definitely share some traits. They tend to be packaged similarly except for the long back. Strong hind-quarters with a good shoulder and balanced neck. Also in the 15h range.
Two cautions: Specifically bred “sport horse Morgans” and colored Morgans. First, I have maybe seen a picture of one maybe two colored Morgans that looked correct. They fall to the general demise of colored horse where the breeder breeds for color and only color. I have seem more club footed, bad shouldered, longed backed, parrot mouthed colored Morgans than I care to count.
The sport horse bred Morgans can occasionally really stray from the Morgan standard. Many have no type to them.
As to personality, they are amazing. They are very, very intelligent, kind, and extremely sensible. I often think the reason I don’t get along well with TB’s is because I grew up with Morgans. I have no patience for a horse that is a “stupid spooker” as I like to call it. Generally a Morgan will assess the situation and then make a decision. They are not a breed to jump and run before asking questions. Of course there are always exceptions and the nature versus nurture; but that has been my personal experience.
They are not necessarily and easy horse. They have an opinion. And when they learn something, they can like to show off…What do you mean don’t passage now? Look how good I am at it! 
The hard thing with the height is many of the over 16h Morgans loose their type. Which is why I don’t think you see a lot of them. However, for starters I would recommend looking at Meadowair Farm in Walpole, NH owned by Andrea Chickering. She has always bred correct nice horse on the big side. They tend to have EXCELLENT minds and be easy going.
On another front Scot Tolman in Spofford, NH breeds the occasional Morgan Dutch cross. With his Dutch horses aimed at Dressage this could be a good venue for you to explore. I myself am waiting on my Morgan mare’s first foal by a Dutch Stallion 
I have a feeling I will always own a Morgan. Though the years I’ve had various h/j and dressage trainers try to take me in other directions and I always end up with a Morgan
I’ve had my mare for 8 years and love her (most of the time!). She is a bit of a princess but I let her be that way. I’ve never connected to another horse like her. I can point at her to move a leg, yet she still occasionally takes off with me to jumps! Trainers tend to hate her as she is very hot and sensitive but any barn worker she’s met she’s brought to her side! The last barn she was at she convinced the manager to let her have free access to the indoor and her stall or her stall and a private grass pasture so she could decide if she wanted to be outside. The manager had her own horses there and yet my mare got all the special treatment 
But take my words with a grain of salt as you can tell how biased I am!