Rate of convergence vs Order of convergence

Question

Consider a numerical scheme converging to the weak solution of the following initial value problem,

\begin{eqnarray} u_t+f(u)_x=0\\ u(x,0)=u_0(x) \end{eqnarray}

There are two important parameters which indicate, how good a numerical scheme is.. Namely rate of convergence and order of convergence..

I roughly understand that first one indicates how fast the scheme converges to the weak solution and the second one indicates how nicely the scheme captures smooth solution..

I have the following doubts..

1. How exactly these quantities are calculated for a given numerical scheme?
2. Are these two independent quantities? (Do we have something like if rate of convergence of a scheme 'A' is more than some other scheme say 'B', then can we say order of convergence of scheme 'A' is more than order of convergence of 'B' and vice-versa )
3. Like higher order schemes, is it possible to construct schemes which have higher rate of convergence?
4. Does the rate of convergence depend on the initial data? Does the order of convergence depend on the initial data? (for what type of initial data rate and order are maximum)
5. Does the nature of flux affect rate and order of convergence... (For example if we use a Lax Friedrich scheme for different types of fluxes keeping the initial data fixed how will these quantities change? is there a specific trend?)
6. They say so called 'Monotone schemes are at-most first order accurate', is there an explicit example where scheme is of first order and where order is strictly less than one.

I read few books on conservation laws but could not get the answer to these questions..

Any guidance would be appreciated.

Definitions:

i) Rate of convergence of the scheme is defined as the rate of convergence of the error $\| u(.,t)-u^h(.,t)\|_{L^1}$ as $h \rightarrow 0,$ where $u$ is the solution and $u^h$ is the numerical approximation at time $t$.

ii) Order of convergence or order of accuracy of the scheme is the largest $p\geq 0$ such that for any smooth solution $u$ the following holds: $$u(x,t+ \Delta t)-H(u(x-k \Delta x,t),\dots ,u(x+k\Delta x,t))= \mathbb{o}(\Delta t^{p+1}).$$