Do the Pokemon you put in Pokemon Dream World, affect the Dream World in any way?
Yes, as awesomesocks42 mentions the type of Pokemon you put to sleep influences which area of the Dream World you will visit.
Here is a list to help you determine which type will increase your chances to visit a particular area:
Q: Does the type of Pokemon I tuck-in influence which area I go to? A: Yes, depending on the Pokemon's type you'll visit a certain area more often than others, here is a list of which types each area prefers (list is tentative):
Type - Area
Bug - Pleasant Forest
Dark - Spooky Manor
Dragon - No Preference
Electric - Pleasant Forest
Fire - Rugged Mountain
Fighting - Rugged Mountain
Flying - Windswept Sky
Ghost - Spooky Manor
Grass - Pleasant Forest
Ground - Icy Cave and Rugged Mountain
Ice - Icy Cave
Normal - Dream Park
Rock - Icy Cave and Rugged Mountain
Poison - Pleasant Forest
Psychic - Spooky Manor
Steel - Icy Cave and Rugged Mountain
Water - Sparkling Sea
yes, they do, as of the global link redesign. I am yet to discover exactly which areas are accessed more by which pokemon, but there is now a correlation, for instance, tucking in a flying type will nearly always result in going to the windswept sky, and tucking in an ice type will nearly always result in going to the icy cave.