So I actually have a good deal of knowledge on post-Roman Medieval Europe, particularly concerning the Visigoths of Spain. However, I am less knowledgeable about the linguistics portion, though I believe it has to do with the size of the invading Germanic peoples versus the much larger local population and Latin's use by the church.

It depends on which group you're talking about. Each invading group of Germanic peoples left their own mark upon the territory they conquered, yet the influence the have on those regions today depends on their integration with the local population, and more importantly, how long the actually asserted authority over that region. For example, the Ostrogoths who conquered Italy were Arians who remained separated from the largely Catholic Roman population and only ruled Italy for perhaps 60 years before succumbing to a Byzantine invasion and effectively being destroyed as a people; thus, their modern day impact is negligible. A similar story exists for the Vandals of North AFrica.

As far as the Visigoths of Spain go, in the ~250 years the Goths ruled before the Islamic conquest they undertook a very slow form of Romanization and integration. At first, the Goths had a separate code of laws, had a different religion(Arianism) and spoke their own language. The language was the first to go, as doing business with the natives required Latin and the church operated in Latin. These reasons were probably why Germanic languages never caught on in the predominantly Latin speaking regions of the former Roman Empire. Visigothic kings during the 6th century tried to keep the Goths and Romans separate, and had strict laws surrounding marriage and religion designed to preserve Gothic identity. Arianism was seen as a Gothic religion and Catholicism was Roman. Leovigild was really the last true Visigothic king in the traditional sense. His son Reccared had a change of policy and around 600 converted the kingdom to Catholicism and removed the separate laws, effectively integrating the population. Even so, Gothic influences remained and can be still seen today in the form of names and loan words, styles of art, and other cultural aspects. Even after the Islamic conquest the Kingdom of Asturias is believed to have been founded by a Gothic nobleman, helping preserve the Gothic influence on the larger dominant Roman culture.

The story of the Franks and Lombards are a bit less clear cut than that of the Goths due to hazy beginnings, and are also considerably longer and more complex (not to mention I'm not as knowledgeable on them), so I will pass on explaining them, though it is doubtless that they both have had enormous impact upon the regions which they ruled, taken that France is a name derived from Francia, or land of the Franks. It is also of note that the Frankish kingdom was the predecessor to both France and the Holy Roman Empire, so they did not entirely separate from their Germanic roots. But alas, these are big questions that will require an amount of self-study to fully grasp the nature of these medieval kingdoms.

Sauces: The Goths in Spain, Thompson. The History of the Medieval World, Bauer. From Roman to Merovingian Gaul: A Reader, Murray.

I'm not qualified to cover the question about language but I can give you some help with the first two questions.

First we have to understand what is meant by Volkerwanderung--the typical way it is understood is the movement of Germans into Roman territories but it goes much further than that: the entire span of the period sees not only the Germans but also people from what we understand to be Central Asia moving into Europe and parts of East and South Asia.

Now we have to understand WHY they began to migrate and invade. There's not really one clear answer to this and many scholars have given their contentions on the matter, but there are two I personally agree with. Chris Beckwith notes that at this point in time Germany resides within a geographic sphere called Inner Asia, and that for much of history the cultures in these regions are based on what he calls the Comitatus--a band of warriors that protect their lord, and in return receive jewels, land, money, precious objects, etc. These articles are more easily and more bountifully, and most importantly they are more easily obtained for longer lengths of time, via trade instead of war.

The Silk Road is essentially the core of the Inner Asian economy, and when Han China closes off their end, it sends ripple effects across the continent, which (in addition to food shortages and some climate change) initiate the Wandering (according to Beckwith). The Comitatus needs goods to keep itself healthy, which necessitates that they move into the regional empires in order to obtain markets for trade. This is precisely what you see early on--elite transfer into the empires.

Etienne de Vaissire is a French scholar who focuses on Sogdia, in what is now Central Asia. The Sogdians are known as great merchants, and in the wake of the Hunnic invasions, it's observed that they move into the regions around the Black Sea rapidly, setting up trade deals. It is also noted that they apparently have dealings with the Huns, and that the Huns are adamant about setting up trade relations with the Romans. A theory he and others have been working on is that the Sogdians sent the Huns into Europe to clear the way (this is in addition to Hun revenge for Gothic attacks) of tribes and to clear out markets. Essentially, the Huns' movement drives other Germans into the Roman Empire, this time it is full families and clans.

So, how does this movement impact the Roman Empire? People are certainly moving into these areas, and the Germans that move in are already somewhat "Romanized", but they also bring their own cultural aspects. By far one of the more important results of this is the desire of the Germans to actually maintain some Roman institutions and culture--the results of a desire to maintain the "Idea of Rome" by the Germans leads to the eventual creation of cultures that we now understand to be France, Spain, Germany, England, etc.

For short term effects, a very noticeable one is the integration of Germans and others into the Roman military, but also the development of walled towns to keep them out, and in order to stabilize the Empire Emperor Diocletian begins to crack down on the citizens, ordering the people to stay put. This is one of the things adopted and kept in place by the Germans--I don't really want to use this term but for sake of conversation this leads eventually to "feudalism" in Medieval Europe.

As for how they're regarded, it's a mixed bag. They're disliked and they're liked initially (depends on the person doing the judging), but as the cultures blend into Medieval Europe, the small percentage of migrants (small compared to the overall size of the Roman population) begin to mix pretty well.

Sources: Empires of the Silk Road, Beckwith. The World of Late Antiquity, Brown. Empires and Barbarians, Heather. The Fall of Rome, Heather. Huns et Xiongnu, Vaissire. Huns and Xiongnu: A New Thought on an Old Problem, Attwood. The Cambridge History of Inner Asia Vol. I.