You've a couple of questions going. Wood is a great floor as it is cheap, non-skid, absorbent (good & bad), and very easily repaired. It can have blocks nailed down to secure items from shifting. I don't think a few pounds of weight is worth losing on an otherwise cheap trailer for the advantages of wooden floors.
Weight is not a problem for the TV, per se, it is the lack of brakes in a sub-2500-lb trailer. I'd upsize to a trailer with brakes as the combination, done well, can stop faster than the car alone. Same for upgrading to tandem axles and shock absorbers. Both make for easier life, IMO. In other words, I'd investigate from a wider perspective a little longer.
Yes, winds will push your trailer around . . unless you load it correctly and keep 12-15% of the total weight on the tongue. Trailer loading diagrams (height and relation to axle forward) are your friend.
Have a look at the U-Haul website, inside and out. And then go to the biggest dealer in your area and look over their trailers.
How much the empty trailer weighs is not nearly as important as handling and braking qualities. And cargo securement.
1990 35' Silver Streak Sterling
; 9k GVWR.
2004 DODGE Cummins 305/555; 6-manual; 9k GVWR.
Hensley Arrow. 10-cpm solo, 18-cpm towing
Sold: Silver Streak Model 3411