Spain could have not entered the Muslim world, it could have remained Germanic. After all it wasn't just the Muslims that defeated Roderik, but the Witizians allied with them. The witizians opened the peninsule for Tariq, and afterwards the only resistence came only from Roderik's party.
So you only need to get rid of this "problem", and the muslims wouldn't have been able to cross the strait.
The loss of the "frankishness" of France was also an internal problem, i know not enough of the matter, but i don't figure history could have easily taken other path.
For Italy... i figure that it would be key that the Visigoths could keep their half of Provence (the Ostrogoths had the other half), against the Franks. Maybe have a king of Toulouse convert from Arianism to Catholicism, so the Galo-Romans wouldn't side almost wholly with the Franks. Franks would still be able to take a good chunk of the Gallic part of the Kingdom of Toulouse, and the Goths might still need to take the capital to Toledo, but the Septimania would be far larger and better entrenched.
Why doing this? Both Gothic kingdoms (visi- and ostro-) were on very good terms and on the brink of unification at times. A strong presence of the Visigoths in southern France would have served as a fall-back for the Ostrogoth independent kingdom after Justinian had taken most of Italy. This would prove useful later, when the Eastern Romans would begin their retreat after Justinian's death. (This is very inaccurate, probably, maybe even ASB: my knowledge of the area at that time is very small, but enough to allow me to fantasize a bit
)
My knowldege on how to stop the slavic expansion, though, is next to zero.
