I think that three days is too little time to go to Romania.

I know 3 days is not much, but I would love to have a short visit to the Transylvania region in Romania and skip Bucharest altogther.
The medieval towns around that region and the Peles Castle is really captivating.

Captivating in the summer, but in the middle of winter?

Your plan is actually pretty good, Brasov and the area around is interesting even in middle of winter. It will be cold (below 5°C/40°F in the first part of the month) but (if there's snow) you'll also get to see scenic winter mountain landscapes.

So fly to Bucharest then head to Brasov - if you want to skip Bucharest completely you could try to catch the minibuses going to Brasov on the road in front of the airport (the airport is north of Bucharest, on the main route to Brasov), although conditions aboard aren't so great.
If you start from Bucharest or the airport earlier in the day you should do Peles palace first, it's located in Sinaia on the main road/railway between Bucharest and Brasov,
If not, go straight to Brasov and do Peles on the last day on the way back to Bucharest.

In the remaining ~2 days do a day trip from Brasov to Sibiu or Sighisoara and another one to Bran castle plus Rasnov fortress.

I would do what Flat4 suggested in the second option "go straight to Brasov and do Peles on the last day on the way back to Bucharest.
In the remaining ~2 days do a day trip from Brasov to Sibiu or Sighisoara and another one to Bran castle plus Rasnov fortress."

In the last day, take the train from Brasov to Sinaia, leave your luggages at train station there and visit Peles Palace. Return then to train station, take your luggage and take the train to Bucharest.

I don't think I am able to do day trips to Sibiu or Bran castle due to time constraint. Basically my trip at Istanbul ends at Wednesday around noon and I have time until Saturday noon before departing to Istanbul's airport.

My plan at the moment are:

Wednesday : Depart from Istanbul by flight and reaching Bucharest
at around 7:50pm. Overnight stay at Bucharest.
Thursday : A quick stroll at Bucharest before departing for
Brasov in the morning. Check in hotel at around 1-2pm
and proceed to Sighisoroa for a short visit before
returning to Brasov late in the evening.
Friday : Explore Brasov until 2pm and of to Peles Castle. Return
to Bucharest late in the evening for an overnight stay.
Saturday : Depart Bucharest in the morning by flight back to
Istanbul.

I can do some minor alteration such as flying to Bucharest on early Thursday morning and spend the night at Istanbul on Wednesday. I know it is challenging and probably kind of tiring to visit all those places in limited time, but those are the essential place that I wish to cover.

Is my plan feasible? Should I drop Sighisoroa or just opt for Bulgaria (Sofia) instead?

What I do is get driven from Bucharest's Otopeni airport to Ploiesti, where northbound trains stop 40 mins or so after leaving Bucharest. The drive to Ploiesti station takes about as long as to Bucharest's Gara de Nord.

Means you can get a later evening train to Brasov (it's less than from Ploiesti), and overnight there. There are departures till 2248. Check times easiest at bahn.de.

It's certainly worth spending just a day or two in Romania. I've been visiting Romania for two days or less at a time for much of the past 20 years. I think my record's 2 hours.

I meant to type:

What I do is get driven from Bucharest's Otopeni airport to Ploiesti, where northbound trains stop 40 mins or so after leaving Bucharest. The drive to Ploiesti station takes about as long as to Bucharest's Gara de Nord.

Means you can get a later evening train to Brasov (it's less than 2 hours from Ploiesti), and overnight there. There are departures till 2248. Check times easiest at bahn.de.

It's certainly worth spending just a day or two in Romania. I've been visiting Romania for two days or less at a time for much of the past 20 years. I think my record's 2 hours.

Your plan is relatively feasible, although you wouldn't really have much (daylight) time to see Sighisoara - if you leave from Brasov after 1-2PM you'll get to Sighisoara around 4PM (in February the sun sets around 5.30-6PM)

It would be best to try to get to Brasov right on Wednesday night, spend the night there and get an early start next morning for the trip to Sighisoara.

My concerns on getting straight to Brasov on Wednesday night is that I will be arriving in Bucharest late in the evening and worry that I might not be able to navigate through the busses or train effectively late in the evening and running on the risk of boarding a wrong bus or train perhaps.

I am not sure if Romania is a tourist friendly country where you can communicate easily in English or if bus/train/airport have signs or information are available readily in English.

The only places that I have been travelling is to are Western Europe and Scandinavian which are more tourist friendly and less challenging as compare to travelling to South Eastern Europe.