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.
Yes
Yes, if they never succeed on their saving throw, and the caster maintains concentration, they would be paralyzed for 10 turns.
Yes, there is nothing preventing a previously paralyzed creature from becoming paralyzed again so it could be done repeatedly.
However failing the save that many times in a row is unlikely. The table below shows the dice roll required to save on the left versus the number of rounds elapsed along the top. The percentages are the odds of having made the save by the time that round is over. As you can see even for creatures with abysmal saves staying paralyzed for 10 rounds is unlikely.
Save Rounds elapsed

Best Answer
Yes, they can be moved
The effects you describe do not force the creature to move on its own, but impose an outside force and push it (pulling would work similarly to pushing). These are two separate kinds of "forced movement", the former usually involving mind control or fear effects and the latter involving physical forces, whether summoned, evoked or mundane.
The description of a shove (PHB 195-6, emphasis mine)
Gust says (XGtE 157, emphasis mine)
Contrast this with a spell like dissonant whispers (PHB 234, emphasis mine)
or the "Flee" option of command (PHB 223, emphasis mine)
A paralyzed creature has a 0 movement speed and no actions available (incl. bonus or reactions) and thus cannot be forced to use them. Even if the spell would grant any of these, the condition would override that effect unless otherwise stated.
Note that the verb choice can be sometimes misleading. The verb "move" can be used both in a transitive (separate actor and subject), as well as an intransitive (actor and subject are the same) way. The above examples use it in an intransitive manner. An example of the transitive use would be the Grasp of Hadar eldritch invocation (XGtE 57, emphasis mine):
In this case the creature performing the move is separate from the one suffering it, and it is thus an outside force that will work on even a paralyzed target.