Difference between differencing data and removing trend line for stationarity In reference to making data stationary for Arima: 
Is there a difference between subtracting a best fit line from data, and a first order difference? 
Or, subtracting an exponential fit from data, and performing a second difference operation? 
Also, if I decompose data by subtracting a best fit line (trend), say I then find the frequency of a seasonal pattern. Say I split the data into columns according to that seasonal frequency, and calculate and subtract the mean of each column. I think this is a way to elliminate the seasonal pattern from the data. 
Is this the same as a seasonal difference operation in seasonal Arima (S-Arima)? 
Should I do these operations before performing an Arima, or are the difference operations in Arima and S-Arima meant to produce the same effect? 
 A: s there a difference between subtracting a best fit line from data, and a first order difference? YES .. actually do this and you will get different results . Your real question is How do I decide which approach to take ?
Or, subtracting an exponential fit from data, and performing a second difference operation?
same answer as above ...
Also, if I decompose data by subtracting a best fit line (trend), say I then find the frequency of a seasonal pattern. Say I split the data into columns according to that seasonal frequency, and calculate and subtract the mean of each column. I think this is a way to elliminate the seasonal pattern from the data.
What I think you are doing here is adding SEASONAL DUMMIES as input series to an SARMAX model https://autobox.com/pdfs/SARMAX.pdf
Is this the same as a seasonal difference operation in seasonal Arima (S-Arima)?
no . The approach you detailed here is to allow for DETERMINISTIC SEASONAL STRUCTURE as compared to SEASONAL DIFFERENCING. Again your real question is which one do i choose and how do I choose it ?
TOO VAGUE TO EVEN TRY TO ANSWER .
Should I do these operations before performing an Arima, or are the difference operations in Arima and S-Arima meant to produce the same effect?
The approach to resolving these opportunities requires 1) a data set and more importantly software/procedures to approach this problem in a holistic way where alternative approaches are tried and evaluated culminating a best/optimal strategy/model for this data set. In other words you customize the solution to the data as the data knows what is going on .. you just have to allow it to talk (so to speak !  .
https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf details this tour de force which can be programmed in many different languages ..it just might take some time .
By the way I hope my New York Approach ( i.e. straight/honest and direct) doesn't betray my roots ....
