Right account for local purchases, loan EMI, and investments

What is the best and most economical way for me to pay the loan EMIs directly? (whether from a Singapore account or a NRE/NRO account)

It is advisable to have it via the NRE account as this would be easier. If you already have funds in NRO account, you can use that before you use the funds from NRE account.

For all expenses I make in India (e.g shopping, general expenses in India visits) what account should I be using, ideally? Is the route to transfer into NRE then NRO and then withdraw from NRO?

Whatever is convenient. Both are fine.

If I plan to make any investments in SIPs/Stock markets, should I link my NRE account with a demat account and directly use that? If I sell the shares will the earnings come back into NRO or NRE?

You need to open a DEMAT PINS Account and link it to NRE account. You are sell and repatriate the funds without any issue from PINS account.

Related question Indian Demat account