It's true than Hunnic emergence in late IVth century Barbaricum led directly or indirectly several Barbarian groups to cross the limes but it was already happening at this point while it was accelerated : it's not even a given that Tervingi Goths that migrated in Moesia in 376 primarily moved because of Huns. It's obviously not the case with Alemmanic presence in Gaul in the 350's.

When Huns appeared in Europe, they did so in the Pontic Steppe within a multi-ethnic region : Sarmatians, Germans, Proto-Balto-Slavs, etc. Usually these tended to blend together in broad coalitions formed around a particularily powerful group. In this case Tanaite Alans and Greuthingi Goths. Trough raid and conflict, Huns imposed themselves in the European Pontic Steppe and while some groups moved, many remained into Hunnic obedience not unlike what happened later with Avars. Until the Vth century, Huns remained mostly centered in eastern Ukraine, raiding Romania and Persia trough Caucasus and while they might have caused a more or less important ruckus in central Europe with groups moving out of their reach (Greuthingi, Trevingi, Taifales, Sarmatians, Vandali, Quadi) which merged, fought, formed their own coalition (such as Radagast's), they didn't do so under Hunnic orders. You'd notice that at this point, we're talking of the broad Danubian region and that Germania as we understand it isn't concerned.
Even at this point, Hunnic political importance wasn't based on a state or imperial base, but on a broad hegemony over submitted but more or less importantly submitted peoples, diversely understood as a "stateless empire" or "super-complex chiefdom", an embedding of peoples. It wouldn't differ much by the Vth.

Still, the Vth century saw several changes in Barbaricum. Other, new, Hunnic groups entered in Europe and Huns advanced south to the Danubian limes, founding new regional hegemonies such as Uldin's on the Lower Danube, provoking further migrations from their neighbours. Their presence along the Roman limes, however, had another consequence : as other Barbarians, they entered in Roman military service, especially to help counter other Barbarian's migrations their presence caused in first place (Hun auxiliaries played an important part defeating Radagast for instance). This payed service, plus the imperial subsides they obtained from Constantinople allowed Hunnic chiefs (at this point importantly Sarmatized and Germanized, with proper Hunnic elements being probably the minority) to increase their capacity of both social and military mobilisation from one hand, and their capacity of redistribution biding subordinated people together, forming a roughly unified hegemony. It was still largely an indefinite power and a previously dominated people could still go their own way as Alans did in the 410's. Again, not an empire and especially not one stretching from Danube to Baltic; but an hegemony over Pontic-Pannonian peoples.
Again, at this point, Huns are absent from Germania, altough them being more present in the Danubian basin and the Pannonian Plain partly led Vandalic and Suevic poples to move in Gaul in the late 400's (which Huns were payed to fight against).

Such a militarized coalition wasn't unconsequential or feeble, tough, and the unified leadership of Attilla certainly points that. But he wasn't that of a conqueror in the sense it appears he wanted to takeover Romania or part of it, than searching first to extract subsides and tributes from the emperors to fuel his hegemony over Barbarian groups. Thing is, while he successfully raided Balkans and Danubian basin, this source dried out as he couldn't bypass Constantinople and as the eastern Empire could focus anew against him. Raiding the western regions were the obvious alternative : it's trough organizing these raids that Huns managed to gather some groups in Germania under their leadership and not trough outright conquest. Hunnic prestige (having defeated Romans diplomatically or militarily) played an immense role there (as it did for Goths). Did Attilla at this point hoped to outright conquer parts of Gaul or Italy from the Empire? This is possible, but it's equally (if not more) possible he wanted to take part in western imperial politics following the Gothic example trough a mix of raiding and service (initially, Attila had proposed an alliance with Romans against Goths).

As it failed due to a Gothic-Roman uneasy alliance (each side willing to set aside their own fight for regional hegemony in the West against a third challenger), however it mostly went downhill : the death of Attilla who binded the hegemony together thanks to his prestige, charisma and leadership allowed several groups to splinter or to make a go at succeeding him, either Attilla's sons, Ermanaric or Ardaric. The nature of such super-coalitions made it vulnerable to change of fortunes. Huns survived as they did in the Vth, splintering in various groups or part of various groups (as possibly hinted by a western practice of cranial deformation in Europe).

Eventually, while Hunnic hegemony had a main role shaping the late IVth and early Vth century, with lasting consequences, it wasn't an empire and didn't structurally challenged how "things were done", so to speak, in Barbaricum. Their structural influence in Germania proper is virtually absent : the development of transrhenan Germania is more attributable to later Merovingian period and especially Carolingian period with a strong effort of Frankish settlement and territorial organization.

- The Fall of Rome, Peter Heather
- Les Huns, le grand empire barbare d'Europe IVè-Vè siècle, Istvan Bona
- Le Dossier Attilla, Iaroslav Lebendynsky

Because the Romans weren't defeated by the Germans - they largely invited them into Rome and made them somewhat Roman. Germans later advanced through the ranks of the Roman vassalage system and military and participated alongside Romans in a series of civil wars that tore the empire apart. Modern historians have pointed out that the "German" warlords that destroyed the Roman Empire were culturally very Roman.

The Huns were in no way militarily superior to the forces of the Western Roman Empire. This was proven in their defeat at Chalons, which at the very least indicated that the Romans were able to fight them to a standstill.

The Romans needed barbarians after a century of intermittent civil wars to repopulate the Empire. The first Germans were settled on German soil by Emperor Constantine around 310 AD. Afterwards, Germanic settlement increased. Germans rose through the ranks of the Roman military as any other nationality in the empire had the ability to do. By 400 AD, Flavius Stilicho, a half-Vandal, was appointed by Theodosius as the de facto regent of the Western Roman Empire. Germanic representation in Roman politics increased in the 400s and onwards and was greater than the representation of any other group within the Empire (Rome was a highly multi-ethnic and multi-cultural society) except Italians. This owed to high representation of Germans within Roman military structures, at a time when Rome was ruled by generals.

These German officials were no less rebellious than those hailing from other backgrounds. Constantine III, an ethnic Briton, proclaimed himself Emperor in 407, attempting to usurp Honorius. He was opposed by several loyal Germanic generals serving under Stilicho. In the late 460s, Riothamus of the Britons, a loyal vassal of Rome, fought to suppress a revolt in Gaul, but was betrayed by Arvandus, an ethnic Gaul who served as prefect of Gaul. In the 460s, Aegidius, hailing from an ancient aristocratic family, ruled Northern Gaul independently from Rome when he disagreed with the policies of the imperial court.

In 475, the Italian Julius Nepos invaded the Roman court on behalf of the Eastern Emperor to depose regent Gundobad. He was overthrown by another Italian, Orestes, an Italian aristocrat, who installed his son Romulus Augustus as leader. Orestes was then overthrown by the "Germanic" Odovacer, who was born and raised in Rome and served his entire adult life in the Roman army. When Odovacer proclaimed himself King of Italy and dissolved the Western Empire, he swore allegiance as a vassal of the East and maintained the Senate. Most of the other "barbarian" warlords also swore allegiance to the East, while the native Sygarius, son of Aegidius, did not.

This gets to the heart of your question. The Romans were not militarily deficient or incapable of holding out invading tribes. Widespread labor shortages gave Rome no choice but to incorporate Germans into their society, either as vassals or as citizens. However, the history of Rome since the late Republic was one of incessant civil wars, with several interludes. Germans, once they gained high offices, just happened to participate in these wars as well.

To answer your second question, Romanized institutions and religion spread Eastward through the Frankish Empire. Sygarius, who I mentioned earlier, was defeated by Clovis of the Franks. Clovis's descendants were less effective, and their "mayors of the court", or chief ministers, grew more powerful. Court mayor Charles Martel defeated the Muslims at Tours and won a civil war which granted him de facto control over the Frankish Kingdom. His descendants Pepin and Charlemagne would spread the Frankish Empire to the East, forcibly converting the Germans to Catholic Christianity and imposing an imperial system of administration upon them.

Sources:

Hughes, Ian. Gaiseric: The Vandal Who Destroyed Rome.

Jordanes. De origine actibusque Getarum.

Bowersock, G.W. The Vanishing Paradigm of the Fall of Rome

Effros, Bonnie. The Enduring Attraction of the Pirenne Thesis

Okamura, Lawrence. Romans, Barbarians, and the Transformation of the Roman World: Cultural Interaction and the Creation of Identity in Late Antiquity.

Kingdoms Of The Empire The Integration Of Barbarians In Late Antiquity