The usage stats from the British National Corpus (BNC) and the Corpus of Contemporary American English look as follows:
speeded.[v*] 259 149
sped.[v*] 1607 302
So sped is preferred over speeded on both sides of the pond, though considerably more so in the US. The interesting part is this, however:
speeded up 178 139
sped up 324 8
That is, when it comes to the phrasal verb to speed up, the preference is not anywhere as strong in the US, and is outright reversed in the UK.
As to usage over time, the Corpus of Historical American English paints the following picture:
(X axis: year, Y axis: incidences per million words.)
So sped has been preferred over speeded for as long as the corpus data goes back.
Generally speaking, irregular verbs tend to become regular over time, rather than the other way round, though the latter is not unheard of, either. However, the more heavily used an irregular verb is, the less likely it is to change. (That is true of other irregular words, too—for example, you won't see childs superseding children any time soon.)
Dived vs. dove has been discussed elsewhere on this site. See also these related questions: