Lightning network: funds into channels Is there a minimum limit of bitcoin that can be added into a channel during the initialization? 

And, if I decided to open a channel with a lightning node, also that node has to add money to the channel?
Is there a minimum limit of bitcoin that can be added into a channel
  during the initialization?


According to the Lightning Network Specifications: 

"The amounts for each output MUST be rounded down to whole satoshis. If this amount, minus the fees for the HTLC transaction, is less than the dust_limit_satoshis set by the owner of the commitment transaction, the output MUST NOT be produced (thus the funds add to fees)."

So, the commitment must be greater than the sum of the transaction fee (determined by miners at the time of broadcast) and the dust_limit_satoshis value. The dust limit purpose is "to prevent creation of uneconomic outputs (which would otherwise remain forever, unspent on the blockchain." - BOLT #5


  And, if I decided to open a channel with a lightning node, also that
  node has to add money to the channel?


From what I understand, this is not typical, but it depends on your agreement with that other node. "Initially they each hold a bitcoin transaction that sends all the bitcoin (e.g. 0.1 bitcoin) back to one party." Lightning RFC - Introductions. So, if the fundee needed to make a payment to the funder, it could provide a payment up front.