My understanding is that technically, B-S uses the 'short rate' which is the instantaneous rate of borrow/lending for term T, denoted $r_t(T)$. I.e. at time $t=0$, if you invest £1 risk-free for term T, at T your investment will be worth $1\times e^{r_{t=0}(T)*T}$. Now, to obtain values for $r_{t}(T)$ you need to construct a yield curve for varying T. Note that $r_t(T)$ isn't directly observable, so we have to calculate discount factors first then convert these to $r_t(T)$.
The first step is to choose which instruments to use in order to do this, addressing your first point. When large banks engage in buying/selling options, shares etc. they will fund this by lending/depositing from other banks at the LIBOR (soon to be SOFR/SONIA) rate, not the treasury rate. Therefore, you should construct what's called the 'money market' curve, i.e. use LIBOR rates for short term (t<3m), interest rate futures for the medium term (3m<t<1y) and interest rate swaps for the long end (1y up to 20/30 years).
So let's say you want the short rate, $r_t(T)$ for term 3 months (T=0.25). First, look up the Libor rate for 3 months, denoted $L_{3m}$. Now if you invest, £1 at $L_{3m}$, in 3 months you will get back $1 + L_{3m}*0.25$. So the discount factor for this period is,
$$
\delta_{3m} = \frac{1}{1 + L_{3m}*0.25}
$$
so now all we have to do is convert this to the short rate $r_t(T)$. This is simply,
$$
r_t(T) = \frac{-1}{T} * ln(\delta_{3m})
$$
(assume now is t=0)
Now by repeating this for the various other LIBOR tenors and doing similar things for the Futures/Swaps you will get a set of values for $r_t(T)$ which you can then interpolate to get an estimate for values in between. At this point you have your yield curve and you can just pick off the $r_t(T)$ for the tenor you are pricing your option for.
This is a rather naive approach to constructing the money market curve in general, however, yield curve construction is a whole field in itself and for your purposes this should be more than enough.
*Note LIBOR/Futures/Swap rates are published daily for various currencies on the CME Group website.