Re: does the size of a piece of fruit affect how many seeds it has inside?

The accurate answer to your question is, no, the size of the fruit does 
not influence (affect) the number of seeds inside.  However, in nature the 
size of the fruit and the number of seeds inside are often related, so 
bigger fruits will contain more seeds, or bigger seeds, but not because of 
the fruit.  This is because plants can use their limited resources 
efficiently.  So fruits containing more developing seeds get more 
resources, so both fruit and seeds become bigger, and smaller fruits with 
fewer seeds get less.  In many cases plants will drop smaller fruit rather 
than wasting their resources.  This sounds as if plants could think about 
such decisions, and we tend to describe these actions as such, but other 
mechanisms are used.   

Fleshy fruits function to attract and reward seed dispersers, and larger 
displays of fruit will function to remove more seeds.  So resources used 
to make bigger fruit containing more seeds result in more offspring being 
dispersed, which is necessary and critical for their survival.  All of 
these relationships are predictions based upon evolutionary theory, and 
all have been demonstrated to operate as predicted.  The biology of plants 
is shaped to result in the successful production of offspring.

The general relationship between fruit size and seeds doesn't hold for 
many domesticated fruits because humans have selected for larger and often 
seedless or few seeded fruits.  Among those grapes that have seeds, the 
biggest will still only have at most only four seeds.  Tomatoes, squash, 
and pumpkins would follow the general rule, and larger fruits will have 
more seed than smaller ones, and you could try counting them, but it would 
take a while.  Wild plums are small and domestic plums are large, but both 
contain a single seed.