So there are many web-wallet services providing sites.
How they program it?
developers must have a background on Blockchain, cryptocurrency and Basic knowledge of dealing with sites and their creation.
html/css - javascript [node.js] - [python/php] - [SQL] - Security team.
Are there bitcoin's opened API to generate wallet and address to receive coin?
for cold wallets.
How about other alt-coin's cases?
You mean support more than a coin?
All of these currencies are the protocols of Blockchain's, success in designing a wallet for the Bitcoin means the possibility of adding any other number of coins.
If that coin is based on bitcoin sources and copied, then they also automatically provide API?
Same above