How did the older games influence new generations?

The core concepts have been the same since Gen 1. Start of game with a Professor, choose name, choose one of three Pokémon, catch and raise Pokémon, travel to 8 gyms, eventually fight Elite 4 and become champion. The differences lies in updated features and new regions, Pokémon and conflicts, but the core has always been the same and I hope it always is, it's an idea that has and can be expanded upon in lots of great ways.

 

The genders from gen 2 affect the future games Pokémon stats.

 

I've never heard of this, what do you mean? It's not possible to transfer Pokémon from Gen 2 into games of later generations because the way Pokémon data was stored changed significantly, so surely you can't mean that the genders will affect the stats given to Pokémon once transferred to a later generation.

 

I think generations were a natural progression and evolution of the series. Gen II introduced forms on unown, gender, happiness, breeding, weather, among other things. Without it a lot of things wouldn't have been the same. Likewise, Gen III introduced abilities, natures, expanded on berries and breeding, and gen IV made the so important physical/special split.

I don't really see them as influences, but rather, just a growth. Without Gen II there'd be no Gen III and without Gen III there would be no Gen IV.

That said, Gen I did influence Gen V a lot, since Gen V was like a series reboot, and it did a pretty good job at that.

 

Pretty much what Slugkid said. The games just build on each other.

 

I feel like breeding creates a whole new strategic aspect in Pokémon. You can breed to get the best nature and great egg moves, but how long will it take? It adds patience to the game, and I think it's great.

 

Each one adds story that builds the wonderful world of Pokémon. They also build memories for us players

 

Every generation (except for V) introduced cross-generation evolutions. Gen IV had 30 cross-gen evo's! That's a big part of the Pokédex, and personally, I love to see cross-gen's, they give older and a bit forgotten Pokémon more attention!
Apart from that, every generation introduced some standard Pokémon (every gen has Starters, a Rattata, either a Pidgey or a Spearow, a Legendary trio, a Pseudo-legend etc.). It's all a repeat of what started in gen I.