How to get the capacity of remote channel in Eclair?
While Eclair itself doesn't provide the information, you can use the information from Eclair to look up the capacity using some other block information provider.

The capacity of channels is not part of the gossip information broadcast over the Lightning Network. The capacities are looked up on the blockchain from the shortChannelId, which fortunately is returned by AllChannels.

The shortChannelID is a triplet consisting of a block height, the index of a transaction within the block, and the index of the output in that transaction. This is the funding transaction outpoint and the amount in the txOut it refers to is the capacity of the channel. Eg,508856x657x0 is this txOutPoint which has a capacity of 0.004BTC.

Perhaps of interest, Blockstream's explorer recognizes the standard shortChannelId encoding &lt;height&gt;x&lt;txIndex&gt;x&lt;outIndex&gt; for searching and will take you directly to the information. I'm not aware of any other block information providers which also do.

Eclair's API doesn't provide you the funding public keys to verify this transaction yourself, but you can probably assume that Eclair has performed the validation else it would not have been accepted into its database.