smog firstly any other help my email is in my profile, we are living just outside the centre to the West. My wife is from Paris so will be able to help with any questions.
£300 for 3 nights in Jan very do-able and you should get something good even spending less than that. Booking.com is good enough certainly for research. Air B&B widely used if you want to go that route (sounds not)
Hotel. We recommend Best-Western at Trocadero (some rooms have Tower view, book direct and make the request) – Trocadero itself has the best elevated view of the tower and a metro station and a good place to start a walk. Fabulous cafe/patessiere Chez Carette. The Shangri-La Hotel is wonderful and possibly may have some deals but maybe out of budget (we are biased as we had our wedding reception there – as above we got a big discount out of season so call and ask inc place of prior poster)
Saint-German very popular with tourists but retains its classic Paris charm, well located (for a nice dinner not too pricey Chez Fernand – share a cafe Gourmand desert). Good area recommendation.
Moulin Rouge area not the best to stay imo, fairly cheap though
Download google maps offline and walk a lot (normal big city caution for pick pockets)
Buy a 10 ticket “carnet” for metro – can be bought in machine
La Forchette / The Fork app for restaurant discounts
Uber works (been on strike recently but that’s France 😉 )
Will send you/post here a walking route suggestion