Upgrading T3 to T4 is simply not very cost effective - you would need to buy new chassis, topdecks, layshafts, shocks - will be much better to just sell off your t3 and get a t4, new or used up to you. I assume the situation will be similar to the TRF's, they are simply different generation cars.
If you are new to TC and running on a small track such as a tennis court (with presumably quite low grip), definitely would not recommend modified (4.5t) motors. You will likely break your car many times over. If you're daring, start at 17.5 level as it seems like you have some rc experience. Complete newbies usually start in the 21.5 or 25.5 levels. 13.5 is very quick also and probably not recommended for starting out.
Never heard of running TC's on 3s batteries except for straight line speed records or something... If you want to go faster just upgrade your motor. With a 4.5t system you will have more than enough power and speed so don't really see the point of this.