Try an op-amp in comparator configuration - nothing in the feedback loop, ground the non-inverting input (or use an offset as needed) and your signal to the inverting input. No feedback. If you use a chip designed as a comparator, like LM319, it won't have internal compensation and is much faster than an equivalent op-amp.
LM319 has full scale response of 80nS (can stretch out to 180 with over-voltage input) which fits your frequency needs. Use capacitor coupled input for zero-crossing detector, or add about half your full scale input to the + pin. LM319 works with single +5 supply but so will a lot of others. Note the old standby LM311 you will see in a lot of examples is too slow.
TL081 is not a particularly good choice for low DC voltages (it's better for AC coupled designs). Precision op-amps have offset voltages in the 10's of microvolts or lower, and there are even auto-zero op-amps that have negligible offset voltage, an drifts in the tens of nV/°C (at some significant cost in other characteristics). It's also quite noisy (25nV/\$ \sqrt{Hz}\$), but at least has a typical flicker noise corner frequency in the 1Hz range. Aside from high input Z, it has basically one really nice advantage, it's really cheap, and widely available, which is why I've actually designed such a 37-year-old op-amp into a new product recently.
A good general purpose (low noise, capable of handling +/-15V supplies, low distorion for AC signals) precision op-amp suitable for mV levels might be the OPA209A.
You can certainly null the offset voltage of your TL081 out using a trimpot as shown on the datasheet, but it won't stay that well nulled for long. A 10°C change will typically change the offset voltage by 100uV, and about one out of every two will be worse (there's no guarantee how much worse, but a guess would be most are better than +/-30uV/°C). An OPA209 is going to be roughly an order of magnitude better.
There are probably going to be better (and many worse) choices for any given application, all things considered. It's amazing what performance you can get for a a few dollars, so it's worth looking around rather than trying to make a silk purse of a sow's ear.
Just to give you an idea of the kind of (in) accuracy you could get, consider that the gain of the TL081 is only guaranteed to be >15,000, so a gain of 1000 amplifier could have a gain error in the 6% range even without the input offset error (which would be very temperature dependent, and has a -3dB corner of something like 20Hz. Cascading two \$\sqrt {1000}\$ gain amplifiers would help with that (null only the first one).
If the range is, say +/-5mV input, frequency is 0.001 to 1Hz and required accuracy 1% of FS + 50uV**, it might be typically *** okay in a lab environment with a light output load, if you null it after warm-up.
** Instrumentation type specification- it means the output could be as much as +/- 100mV from the ideal value with any input, so a 1mV input could give you 900mV or 1100mV.
*** "Typically" means that one chip might be okay, and the next might not meet the requirements. Guaranteed value is probably 10 times worse.
Best Answer
By buffering the shield with each input, it reduces ingress of unwanted signals and reduces the capacitance on the coaxial cable as the shield tracks the center conductor signal. Improved immunity is achieved further with triaxial or double shielded cables.
In the case of coaxial a 1MOhm drain wire provides ground reference to the source to shunt higher impedance low frequency stray fields. This is done on the outer shield for the triaxial cable on the source side.