It's...complicated, so pardon me if I keep it short (I'm not at home right now, and not having my books with me to check it out)
Eventually, it all comes down to Late Imperial structures being modified by events.
Kingship: plural in the merovingian dominions (but aristocrats were excluded), but monistic in Iberia and Italy (where aristocrats could aim to kingship)
I'm not too sure about Ostrogothic Italy, but you had a plural kingship in both Gaul and Spain, akin to imperial collegiality, and similarly based on "fronts". (Often south/east, but not only).
The main difference is partially based on a dynastical continuity in Francia, almost sacred* (at the very least really respected), and a growingly anti-dynastical kingship in Gothia (not a demonstration of unstability, but rather a policy to prevent the rise of too powerful kings).
Both eventually worked along a same line of
truste, with nobility becoming more directly tied up to the royal power (which became really a problem in Spain, as it concerned great landowners nobles, which eventually agreed to not remove the benefits given by the various kings).
I'm not too sure about what you mean by "aristocrats being excluded", sure, they couldn't aim at kingship in Francia proper, but could so in peripherical regions (namely Aquitaine and Bavaria, while admittedly as sub-kingdoms and in the Late Merovingian period)
*That said, the Frankish sacre used later by French kings is directly issued from Gothic rites of power, themselves inspired by Byzantine's)
Ostrogothic kingship is a pit particular, being far more than its Barbaro-Roman counterparts being understood as a patricianship, a governorate for the emperor. But it's a bit hard to systematize with the Romans-Gothic wars.
-Where do the councils in Visigothic Spain come from?
Well, councils on all Latin Christiendom were a political matter, whatever in Spain or Gaul.
The Councils of Toledo only became something else with the IVth and Isidorus of Sevilla.
Originally it was about stabilizing the Visigothic kingship, making it formally elective and more stable, but the wisfhul thinking about making the Council deciding it instead of nobility was moot.
Eventually it became more how Visigothic kingship get legitimized.
-Succession law: royal dynasties in Gaul, election in Iberia/Italy. Is it only a matter of ill lucked kings, or is it connected to the nature of kingship?
You don't have really succession laws : it's more dependent of both uses and role of the kings, essentially military and redistributive for later periods.
It was customary for both Visigoths and Franks to deal with a same prestigious dynasty, but Balthi eventually failed in Spain and weren't replaced by a strong dynasty, in spite of tentatives, while it happened with Peppinids.
-Military: why the Franks were so powerful, while the Visigoths weren't?
I'm not sure Franks were that obviously powerful military. They were so politically by the Vth/VIth century because they weren't Homeans institutionally, which made the Gallo-Roman nobility falling in love with them after the defeat (rather than from the beggining), but military?
Both kingdoms eventually managed to expand (Germania, Galicia) and to fend off outer raids (Basques, Saxons).
One of the main differences was that Visigoths had to directly deal with Byzantines, when the only real intervention in Francia was during Gondovald's revolt and it was really indirect and in alliance with Austrasia. You'd note that Franks didn't do much better against Romans in Italy.
EDIT : Not that you didn't have differences : Visigothic armies usually included domesticity (on the political sense) including slaves; while I don't think it was widely the case for Franks, where freedmen-in-arms played a certain role. The lead around dukes and great nobility seems to have been more ancient in Gothia than in Francia (but on the other hand never really went the way creating new kingdoms, even if it was attempted with the rebellion of Paulus, for exemple).
Also, i know that the Visigoths were rather tolerant with catholic bishops, so i don't consider religion as a weakness factor.
It was, in the early period, when you consider that Roman nobility was partially religious : Desiderii-Salvii were as well counts than bishops. When your legitimacy is partially due to religion, you may prefer an Orthodox leader.
Now, was it the sole factor? I don't think so, as Alaric II's tentative of union prooved, but it provoked a "snowball effect" when Goths were defeated at Vouillé, that only Theodoric's intervention stopped.
Hermenengild's revolt points as well that you had a political base of dissatisfaction with an Homean power.
EDIT : As for Lombards...Maybe you'd have more chances asking LordKalvan, it's not exactly my strong point.
Apparently, the previous nobility (Gothic and Roman) was more or less replaced, at least partially, contrary to what happened in Gaul and Spain. It is possible that they were less integrated to post-Imperial Romania than their predecessors, which may be an explanation to the maintenance of a non continuous dynastical kingship, but on the other hand they were quickly influenced by Romans, would it be before the geopolitical pressures in Northern and Southern Italy.
Eventually Lombard kings seems to have the same military and redistributive function than in contemporary Francia and Gothia. And eventually, I wonder if we're not in face of a similar evolution than in Spain, alltough for different reasons, in favour of an anti-dynastic succession.