is there any problem with charging the leaf only with 120?

There is no adverse effects to using the 110v, trickle-charge cable, other than it takes a very long time to fully charge the car.

I called Nissan LEAF customer service and asked them some questions about battery life and charging. According to this agent they recommend charging at LEVEL 2. Level 1 is for convenience when you don't have 220V. Will trickle hurt it? Don't think so, and I did not press the agent on if it would reduce life. The did offer that fast DC charging is not something you want to do often (if at all). Is there data to back that up or is this their opinion? Did you read the manual. Sometimes there are pearls of wisdom in there.

The issue with the fast charging is it creates heat and the leaf has no ability to deal with heat. Nissan foolishly has not installed any active cooling as most/all other electric car vendors have done. So when you fast charge this creates more heat, which will kill your battery.

Every Leaf on the road has a dying battery just a question of how fast

I think that is true level 2 is best over all. I do believe the fast charge once in a while can help battery so as you watch the temp. I have seen my Ahr and SOH move back up from a proper fast charge.