Does a rogue need to meet the prerequisites of a Combat Feat he or she gains via the Combat Trick talent? Could I, for example, gain Spring Attack without Dodge and Mobility?
A rogue needs to meet all prerequisites for the feats gained via Combat trick.
Take a look at the description of Combat Trick
A rogue that selects this talent gains a bonus combat feat (see Feats).
and compare the rogue talent description to the Monk Bonus feat
A monk need not have any of the prerequisites normally required for these feats to select them.
or the Ranger Combat Style feat
Combat Style Feat (Ex)
He can choose feats from his selected combat style, even if he does not have the normal prerequisites.
Both of these abilities explicitly waive the prerequisites, while Combat Trick doesn't. Since having to qualify for feats is the norm, this enabling (in that it enables you do "go against the rules") text is important.