Quarterly Taxes + Taxes Paid Write Off

[Lyss]

I have 2 tax questions, one about quarterly payment and one about writing off taxes paid.

Question 1:
I just want to clarify if I need to pay quarterly taxes or not. A CPA recently told me (over the phone, without seeing our information) that I need to pay quarterly taxes for "my business" if my self-employment tax is more than $1,000. I believe that next year I may owe just over that in self-employment tax for 2020. Yet after reading on the IRS website, I'm not sure I actually need to do this even if my SE tax is over $1000. I am married and filing jointly. My husband is employed outside of the home and withholds at a higher rate because of my self-employment. We generally, since I've been self-employed, get over 3K back on our federal taxes. Next year I expect to make more and have less big expenses so I'm trying to plan ahead.

Question 2:
Can I write off my state's SE tax as "taxes paid in the prior year" on my federal taxes? Like the amount owed for my SE tax when we file our taxes in February that is taken from our refund (my husband withholds a lot)? The questions that ask about "State balance due paid with 2018 state return in 2019" and "Other state and local income taxes paid" always make me wonder if this is something I should be including.

Thank you!