Is it worth it to earn Bitcoin by mining?

It's a simply thing called hard work, and as long as you are capable of doing it as a matter of sustainability, you can get bitcoins without an investments if you do bitcoin jobs online. One job that earned bigger amounts of bitcoin is the translator jobs or blogging, it can be with other coins but you can trade it to earn good bitcoin amounts.

is any real and trusted site paying bitcoin or cryptocurrency, so that i can earn $2 or $3 per day?

I think there are many ways you can earn bitcoin without the expense of investing, you can join the bitcointalk forum.I think in bitcointalk you can get bitcoin by following the signature campaign, but the ranking is very influential in the payment of signature campaign. Maybe you can join altcoin bounty with your social media account.

Actually in this forum there are many ways how to earn bitcoins, just participated in signature campaign even if the bounty reward its only altcoins but you trade altcoins on bitcoins and have some other signature campaign also the bounty is bitcoin or just try in the services if you have some skills for making logo or banner for there project.

We must earn bitcoin by following various giveaway on this internet, for example is claim faucet, PTC, signature campaign and so on.

There are many ways to earn bitcoin do thats why many people became rich by using this bitcoin. Like signature campaign , twitter , fb campaign it will help to us to earn a lot of bitcoin just do workhard everyday to earn a lot of bitcoin and be patience and have a courage  to use bitcoin.

If you mean a zero cost investment then I suggest you should start doing signature campaign first. From there you will get a profit weekly and you should save them then try to enter trading after you got a decent amount of bitcoins. It is up to you if you want a slow but more effective way of earning bitcoins.

In your question lies the answer and actually one of the most attracting features about cryptos in my opinion.  Bitcoin and alts coins actually make it relatively easy to acquire some coins without investing large sums of fiat. What i have gathered in my experience is that, the more time you dedicate, the more you look and research, the better your knowledge of the crypto world, subsequently the higher the chances of getting in early on new alt coins, signature campaigns, bounties and airdrops.   Admittedly, it is alot easier if you have large sums of fiat to invest in holding or trading, it is not impossible to collect multiple btc without investment.  Research, learn, work smart and earn. 

$0.02 or $0.03 would  already be nice...