Right now I make 100 Doge coins a day from my referrals on the faucets I use.
How long does it take to make 100 Doge with a miner?
Depends on what mining equipment you are owning or going to own.
Should I keep investing in referrals or start mining, that's the thoughts.
Neither. Faucets are dead a long time back. I can say this because I came to the bitcoin world with an interest in how people could farm some faucets or have a passive earning through them. Then Google decided to ban faucet sites from its Ad Network and the faucets became worthless.
But if you are having some good number of referrals then that can be a side earning. I would not suggest mining because Doge in the long run is not a coin worth putting money in, just good for the pump and dump.
Rather using your money to buy bitcoin at low price and selling it for high price is a reasonable investment. It is called trading and specifically long term trading.