Should one have funds in his node to relay transactions?
Having funds in the node's on-chain wallet is not enough. You have to open channels to other nodes and have someone open a channel to you.
Does the total number of your nodes channels depend on your funds?
No. It's up to you with who you will open a channel. Some nodes reject channels under a certain value specified in the config.
Its more difficult than what it seems and as other said not in the spirit of decentralization