How common were independent Counties and Duchies in the 700's to the 1000's?

EDIT: TL;DR That was a bad 300 years for small independent states in Western Europe, although there were independent small kingdoms in England. The Principality of Benevento, which later saw the split of duchies of Salerno, Amalfi, is the best example of an independent Duchy in that time period, which is probably what OP is looking for

I'm afraid you'll have to cast your net wider if you are looking for cool stories of small independent statelets in the middle ages. Although Counts and Dukes could and often would wield lots of autonomous power, examples of true independence are tough to come by in the time period you specify.

Your start date; the 8th century, was a a period where European royalty was particularly busy kicking ass. In modern day France alone, a succession of three Kings made life particularly difficult for independent rulers: Charles "Martel" (whose name literally means Charles the Hammer; he subjugated the Dukes of Neustria, Bourgogne and Aquitaine, cementing modern French borders), Pepin the Short (who was never once defeated in battle), and Charlemagne, who conquered the previously independent "German" kingdoms of Svevia, Bavaria, Thurungia and Saxony as well as the Kingdom of Italy in order to create a massive central european empire. Indeed, the period just prior to your question might be better to look at for stories of independent Kingdoms (Provence, Burgundy and Aquitaine are interesting French examples, while the Lombard Kingdom in Italy is, I believe, by far the coolest).

In the 9th century, interesting examples of small independent states could include the British Heptarchy, or the system of seven or so "Anglo-Saxon" kingdoms in England that formed following the departure of Rome (not including the fascinating Kingdoms of Cornwall, Wales, Galloway, and Proto-Scotland) and the independent Duchy of Brittany, in north-western France. The actual interaction between the various Anglo-Saxon kingdoms in England is a much debated topic, and the very use of the term "Heptarchy" is disputed, so I'll refrain from going into much detail.

On the european continent in the 9th century, although specific conflicts symptomatic of the Carolingan Empire's unwieldily subdivisions did occur (just to give you an idea, after Charlemagne's death, Ludovic, his heir, almost immediately had to face off against his nephew Bernard, who had been given Italy, and promptly claimed he was independent of Imperial authority), and these conflicts did see individual Dukes and Counts sometimes act in conspiracies against their overlords at various levels, unfortunately these occurred within the sphere of the Frankish Empire, and as such don't really answer your question.

However, a good example for you could be found after Carolingan conquest of Italy. A little background: the Kingdom of Italy, ruled by Lombatrds (who by this time had gone native) had always clashed with Byzantine Empire over hegemony in Italy, and under King Desiderious, it looked like Italy could finally be unified; a successful offensive saw the encirclement of the Pentapolis (the last five Byzantine strongholds along the Adriatic Coast that to this day are said to be in the region of Romagna; as the Byzantines were the last "Romans" to leave) allowing Desiderious to march south against Rome unopposed. With the Byzantine army cornered in the Pentapolis and fearing a loss of independence for the bishop of Rome, who had been pretty much guaranteed a free hand to rule the "Duchy of Rome" (modern Lazio) in exchange for nominal Byzantine rule and military access, Pope Adrian cried "Uncle", excommunicated Desiderius, and asked Charlemagne to come to his aid. Charlemagne, busy conquering what would later become Germany, was immediately flattered, and marched for Italy.

Charlemagne marched down from Germany in two columns, one under his personal command taking a westerly route, through the Valley of Susa, and another under his uncle, Bernard, through the Valley of Aosta.

An ancient system of walls called the "Lombard Locks" (Chiuse Lombarde) sealed off the valley of Susa, and defenses personally commanded by King Desiderius held fast against Charlemagne's assault. However, the Valley of Aosta was guarded by Desiderius' son, Adelchi, who had been entrusted with a much smaller force. Pushed back by the enemy at Ivrea, Adelchi was forced into a disorderly retreat. Desiderius withdrew to avoid encirclement, and barricaded himself in the ancient fortress of Pavia, while Adelchi withdrew all the way to Verona. Soon, Desiderius capitulated and would spend the rest of his days in a Belgian monastery; Charlemagne was crowned king of Italy, and Adelchi fled to Costantinople (which, in adherence to the principle, "The enemy of my enemy is my friend" saw the Byzantine emperor welcome the deposed prince now that Charlemagne was claiming the title "King of the Romans").

How does all of this relate to your question? Well, Charlemagne, "King of Italy" had only travelled to Rome; he still had to subjugate the other half of the Italian peninsula. Or did he? Seeing Charlemagne and his army at the gates of Salerno, Duke Arechi of Benevento, a powerful and fiercely loyal vassal of the deposed King of Italy (he had married Desiderius' daughter) came to an agreement with Charlemagne. Sending his son Grimolado as a hostage to Charlemagne's court, he secured the right to continue ruling as Duke in southern Italy; and was even elevated to the rank of Prince.

On the death of his father, Grimolado was allowed to return to Benevento to claim the throne, on the condition that he mint coinage in the name of the Emperor, and demolish the fortifications that dotted Southern Italy. On his arrival, Grimolado immediately began strengthening the fortifications at Benevento and Salerno. Having made his intentions clear to the Carolingans, he then constructed a church dedicated to St. Sophia in Benevento; a direct challenge to the Byzantine empire, just in case they were harboring any thoughts of subjugating him. It looked like the Duchy of Benevento was independent and planning to stay that way.

The next hundred years would mark the ascent of the Principality of Benevento, and perhaps of Lombard culture as a whole; marked by the monuments constructed in Benevento would lead it to be called, "The Second Pavia" (which had been the capital of the previous Kingdom of Italy), and the great wealth of the city of Salerno, which became major mediterranean entrepôt, and become home to a renowned school of Medicine. Successive aggressions, both by Charlemagne's successor Ludovic and his vassals in Italy were successfully repealed, either by force of arms or diplomacy. However, instability would be a hallmark of the principality; as successive invasions by the Byzantine Empire would contest the regions of Calabria and Puglia, while dynastic infighting would split the kingdom three ways by the mid 9th century; the Duchy of Benevento, the Duchy of Salerno, and the Republic of Amalfi (ironically, each capital is within an hour's drive of each other, although the individual territories stretched far in different directions). These three independent states could be examples of what you're looking for; I hope I've answered your question.

I'd suggest Jorg Jarnut's History of the Lombards (Torino, Enaudi, 2002) as (I believe) the most definitive text on the Lombards in Italy, however, I don't know if it's been translated into English. Looking through possible books that cover this time period, the most interesting english text I can find for you is Family Power in Southern Italy: The Duchy of Gaeta and its Neighbours, 850-1139 by Patricia Skinner (Cambridge University Press: 1995). However, I have not read it, I can't tell you if it's any good; it appears to be the story of Gaeta, one of a handful of southern Italian Duchies and Counties that flip-flopped in their allegiance to both the Byzantines and Lombards in a bid to retain as much autonomy as possible between the 9th and 12th centuries.