It's literally almost a complete copy of crush crush. Only with western style and a few extra things.

Obvious parody on fake taxi, so I suppose it's for luring people from porn sites with ads.

And to answer your question - everything you buy in shop carries over aside of the temporary stuff like cash and time skip obviously.

As a free to player I would suggest using the 400 nutaku coins offer(through refs) and then waiting to buy the x8 time warp, it really changes the game from complete stagnation.
Every girl has 10 'levels' first 5 girls give 1 energy for 1st 9 lvls and 20 diamonds on last level. You also get 10 diamonds for free(better not use those when tutorial buttons show you to use it, it's a waste)
So by the time you max 3rd girl with free 40 diamonds you will have enough for time warp which will let you progress with more or less decent speed for a while until getting stuck and grinding for the 2nd x8 warp again, after which the effect of it starts to diminish greatly and you might start saving for 20 energy instead(since it's very limited with only 5 girls having it) unless you plan to dump a lot of money.

The rest of the girls seem to give 1 diamond for level 1 and not sure about anything past that.

But you can't keep progressing by resets unless they suprass previous resets, so in reality -the game bonus is pretty much a fixed bonus, each girl gives progressively more and more per 1 level though.