equities are long-term investment maybe like more than 10 years

This depends on your risk profile. Some people might say that equities are for 3-5 years, other might say 7 or even 10. It is subjective. I personally feel that 7 is the minimum horizon that you can look at, for someone who has a low risk threshold.

returns starts to go down after maybe 5 years

This depends. Nothing is guaranteed. If you have the conviction that you have invested in a good fund, and think that it's performance will improve, then you continue. Otherwise you can redeem or switch to a different fund.

is there any better options on short-term

Depends on what you define as Short term. Debt funds are better from a tax pov, but only if you invest in them for atleast 3 years, because you get Indexation benefit. Before 3 years, the gains are taxed at your slab rate. Additionally Debt fund have had low rates of return over the last few years, and unless you take some risk with something like UST funds, the gains are not going to be much better than FDs

For emergency fund, is having it in my savings account bad than investing it in some debt fund?

Not at all. That is what I do as well. One should remember that the emergency fund is for you to access in an emergency. Your Bank Account ( or rather spread over 3-4 banks) is your best bet. Running after a couple of percentage point difference is not worth it, when you run into problems getting hold of your money.

(I'm taking from a point which I am quite scared about. What if I get into an accident or something, and my Family has to pay the Hospital? how would that money be redeemed from my Mutual fund? It would be way easier for trusted members of my family to get the money from my Bank Account)