Strange behaviour with EllipticTheta functions

Question

For some time I have been using EllipticTheta[3,-,-] functions, and to me it seems that Mathematica is really not handling them in the best way. Below are two of the instances I have faced so far, and I would appreciate any input to resolve these issues.

1. Plots

Here is a plot I have tried and it works well:

Plot3D[EllipticTheta[3, Pi/2  x, t], {x, 0, 2}, {t, 0, 1}]

I understand the peaks as the Theta function approaches a Dirac comb when t=1.

Then I try

Plot3D[EllipticTheta[3, Pi/2 x, E^(- Pi^2 t)],{x,0,2},{t,0,tmax}]

this is how it looks for tmax=1

by increasing tmax things start looking strange, and eventually the kernel seems to crash (from about tmax=4.5). This is how it looks for tmax=5

It becomes even worse for larger values of tmax. I don't seem to understand what is causing the problem, and how choosing larger intervals for the plot messes up the part of the plot for smaller values of t which were looking fine with tmax=1.

2. Derivatives

I have read that derivatives of the Theta function with respect to its third argument are problematic as discussed here, but I do not understand the reason. The solution suggested in that post is to use the corresponding PDE, i.e.

Derivative[0,0,1][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)] = 
Derivative[0,2,0][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)]

So this means instead of Derivative[0,1,1][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)] it's better to use Derivative[0,3,0][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)]. To check, I plot it

Plot3D[ Evaluate[
  Derivative[0, 3, 0][EllipticTheta][3, Pi/2 x, E^(-Pi^2 t)]], {x, 0, 
  2}, {t, 0, 1}]

Note that this is quite slow to generate. Q: How to improve the speed?

In addition, I do not know if the values of these higher-order derivatives are something that I could trust.

Also, when I try increasing t to, say, {t,0,2} Mathematica v12.1 jumps out of the notebook. Note that I am actually trying to use these derivatives as part of an integrand (like here) and I'd want to avoid singularities/numerical instabilities even in the cost of (a bit of) accuracy. So maybe I should use a finite sum, instead of the infinite sum in the definition of EllipticTheta?

Here is another example, which (I think) should be identical to zero (adapted from this post)

Plot[ Evaluate[ 
  D[EllipticTheta[3, x, E^(-Pi^2 t)], {x, 3}] /. x -> 0], {t, 0, 1}]

(edit/update: flinty's solution to define a WorkingPrecision resolves this latter one. I had assumed Mathematica does not need that for built-in functions)

Question

Is there a better way to deal with EllipticTheta in Mathematica? I tried implementing it as the direct summation it is defined through, and ran exactly to the same problems. So I guess Mathematica might be actually evaluating the sum when EllipticTheta is called, and maybe that is causing the problems (since it'll run into exponentials with very large negative powers).