Such a project would be exposed to temperatures of thousands of degrees even while still in the mantle, nowhere near the core.
I'm pretty sure that would lead to many possible modes of structural failure, but probably the electronic systems would fry before it got to that stage.
No. We estimate the core-mantle boundary to be at ~ 3700°K. None of our technology would work at that temperature. The deepest hole we've ever dug is a little over 12 km at which temperatures our equipment failed due (mostly) to temperature (and those temps were far cooler than 1000°C!). For comparison, the mantle-core boundary is at ~ 2900 km. It is impossible to predict whether there will be some "advanced technology" which will allow us to dig down through the Mantle, but the likelihood is vanishingly small based on what we know now.
The pressure would be equally challenging to deal with as the temperature. Whatever the original shape of your "probe" it would be crushed into a sphere with any air gaps being crushed into nothing. I calculated the pressure once and the strength of the pressure vessel material required was far beyond any known material. I don't recall the numbers off hand but calculating the pressure at the center of the earth is an interesting problem.
Stevenson's 2003 paper in Nature illustrates the scale of technology that would be required for such an undertaking.
Nature423, 239-240 (15 May 2003)
Planetary science: Mission to Earth's core — a modest proposal Abstract: Planetary missions have enhanced our understanding of the Solar System and how planets work, but no comparable exploratory effort has been directed towards the Earth's interior, where equally fascinating scientific issues are waiting to be investigated. Here I propose a scheme for a mission to the Earth's core, in which a small communication probe would be conveyed in a huge volume of liquid-iron alloy migrating down to the core along a crack that is propagating under the action of gravity. The grapefruit-sized probe would transmit its findings back to the surface using high-frequency seismic waves sensed by a ground-coupled wave detector. The probe should take about a week to reach the core, and the minimum mass of molten iron required would be 108–1010 kg — or roughly between an hour and a week of Earth's total iron-foundry production.