Was it established that the Cleavers moved to a new home during the 3rd or 4th season or was it supposed to be the same house when they changed the sets?

Yes, it was established that it was a different house. There were a bunch of secondary plots during the 2nd season where Ward and June were going out looking at houses. After the move, there was an episode where Beaver went back to the old house to reclaim a tree that he thought belonged to him.

The actual move was never shown.

well, then,both houses must have been in the same neighborhood since Eddie Hasskel was in the entire series.

Both homes were in Mayfield for sure. The implication is that they were fairly close by, since Beaver was able to walk back to his old house to dig up his tree in 'Beaver's Tree'. Then again, everything in Mayfield seems to be within walking distance! I'd guess that they were probably just a few blocks away from each other.

The real reason for the move is that the production of the series moved from Republic Studios to the Universal lot. If you'll notice, there was a lot more outside activity in episodes after the move- they show was able to take advantage of Universal's large backlot. The backlot at Republic was a lot smaller- both the residential set street with the Cleaver house, and the city set streets. I don't know if the better facilties were a reason for the move, but it wouldn't suprise me.

I notice that a lot of those outdoor scenes must have been filmed early in the morning. You can
often see the actor's breath even though they
are dressed in summer clothes and they never comment on it being cold out.

Both homes were in Mayfield for sure. The implication is that they were fairly close by, since Beaver was able to walk back to his old house to dig up his tree in 'Beaver's Tree'. Then again, everything in Mayfield seems to be within walking distance! I'd guess that they were probably just a few blocks away from each other.

The real reason for the move is that the production of the series moved from Republic Studios to the Universal lot. If you'll notice, there was a lot more outside activity in episodes after the move- they show was able to take advantage of Universal's large backlot. The backlot at Republic was a lot smaller- both the residential set street with the Cleaver house, and the city set streets. I don't know if the better facilties were a reason for the move, but it wouldn't suprise me.
when they moved to the Universal back lot, they used the same house where "MARCUS WELBY, M.D." was filmed later.

The actual move was never shown.

There is a 5-part fanfic, called "Summer of '59" that tries to fill in what happened in making the move: http://www.sitcomsonline.com/boards/showthread.php?t=36939

when they moved to the Universal back lot, they used the same house where "MARCUS WELBY, M.D." was filmed later.

Correct, although the first floor was altered for a very different look. The house was originally built for the 1955 movie "The Desperate Hours" starring Humphrey Bogart.