No. Frame rate issues will NOT be fixed on the PS2. Overall appearance will be improved through the use of bilinear filtering and a higher resolution, but the game will run the exact same way. The reason for that is that the emulated CPU cannot be overclocked, as that would screw up the timing in games. Console games are not designed like PC games, where a timer will simply add frames when the game is running too fast. Console games will update once every CPU cycle in the interest of efficiency, since all the hardware will remain the same through the life of the system. When a game is running on a successor console, the hardware of the previous console must either be emulated, or present in the next console for the games to run, and the specifications of the previous console must be met EXACTLY as they were, or the games won't run right.
The PS2 uses the same CPU as the PSX for the IO controller on the PS2 I believe, and the SPU is exactly the same as the PSX's, it just has more memory. This means that the PS2 hardware actually incorporates the PSX hardware into the PS2 design, and that's what is actually running the game.
The GBA is another good example. It has 2 CPUs, one from a GBC, and one for the GBA. When you put a cart in, a switch is pressed, and the hardware will activate the appropriate CPU for playing the game.
Word has it that the PS3 will not include the PS2 hardware though, and will emulate the hardware through software. This will likely break compatability with a lot of games, but we shall see. Likely they will focus on emulating the popular games properly, and let a few of the lesser known ones slide, so it won't matter much anyways.