D&D 3.5's definition of the paralyzed status reads (emphases mine)
A paralyzed character is frozen in place and unable to move or act. A paralyzed character has effective Dexterity and Strength scores of 0 and is helpless, but can take purely mental actions
So when paralyzed, you can't speak (referring to question #2), as that would be an action, and not a purely mental action. There is so corroborating evidence in the D&D Glossary on WotC's site which refers to the monster action Paralyze:
Paralyzed creatures cannot move, speak, or take any physical actions
though one might consider the fact that it was explicitly added to mean that it differs from the paralyzed condition,
Now, for question #1: a spell can have Verbal, Somatic and Material components. Still Spell only eliminates the Somatic component. Adding Silent Spell will take care of Verbal, and Eschew Materials (for most standard materials)/Ignore Material Components (for the rest) for Material (apart from Focus or Divine Focus, which are much harder, if not impossible, to get rid of), as KRyan and Vereos noted in comments. However, this question still isn't addressed explicitly in the SRD. It is mentioned in the Official FAQ, though:
What exactly counts as a “purely mental” action, such as those allowed to the second mind created by the schism power (XPH 130)? Does a silent, still spell count?
The schism power specifically includes manifesting a power as an example of a purely mental action. Elsewhere in XPH, this is expanded to include “casting spells with no verbal, somatic, or material components.” Thus, a silent, still spell wouldn’t count unless the spell also had no material components (including focuses)
So if you accept the FAQ as RAW, and willing to accept the Expanded Psionics Handbook are RAW for spells, then these three feats would be enough to cast a spell while paralyzed, or any subset of them for spells without the relevant component.