Read my post again - I already answered these
I'll be even more explicit here though, and then explain in more detail.
You can't upgrade the GPU.
You can't move the system to a new case in order to upgrade the GPU
You didn't link a 1070, but it won't fit. "mini" 1070s are not mini in the way you would need them to be mini. Same with other cards with similar names.
Your PC is in a special SFF case. This means GPUs need to have 'short' PCIe brackets to fit. The only cards that have this are the lowest power ones like the GTX 1050 ti.
The 1070 mini's "mini" refers to the GPU length, i.e. how long it is measuring from the back of the case towards the front of the case. Regardless of whether you can find a GTX 1070 (or any other GPU) short enough to fit in your case that way - it won't matter. Because the PCIe brackets will be the wrong size so you will be unable to install it.
Regarding moving the system to a case that would fit the GPU, you won't be able to fit the motherboard and/or connect the case's front panel to the motherboard.
And (according to material via google) the PSU doesn't have a PCIe power connector to power more powerful graphics cards either as far as I can't tell. And you can't replace the PSU because the power connection to the motherboard is proprietary.
I'm reluctant to even mention the next bit in case you think it's a good idea. it's not, it's a really, really stupid idea. However, to prevent someone else posting this and failing to address how dumb it is, I have to cover it. There are a couple of models of GTX 1650 that are technically more powerful than your 1050 ti and that would fit your system. I say technically, because it still won't achieve close to what you are looking for, and is only a bit more powerful than your 1050 ti (despite the number in the card's name looking bigger). You therefore should not buy one of these special 1650s. Your funds are (I'm assuming) limited and spending that kind of money on a very limited upgrade is a terrible return on your investment versus saving up to buy a new system. It will give a bad experience for the money, and only make it take longer to get your inevitable upgrade as you need to save up those $150 again. It'd be like shredding $150.
To specifically address the existing parts you have and reusing them:
- Mobo - no, because proprietary connectors, and possibly irregular form factor
- PSU - same, also inadequate PCIe connectors for better GPUs
- CPU - no, because you need a compatible mobo - and compatible used mobos on ebay can sell for more than new mobos supporting new hardware - and a lot of the cheapest ones on ebay are taken from systems like Dells and HPs that have proprietary connectors. Your CPU will be a limit to performance in some titles, especially with a GPU upgrade, so sinking more cash into re-using a CPU that you will want to replace soon anyway is again a bad use of money.
- RAM - no, because new motherboards use DDR4 and your is DDR3.
- GPU - yes, but that was the part you wanted to upgrade to begin with.
- SSD - yes, you can use the SSD in a new system at least
Sorry that news isn't better but I hope that explains more clearly why what you're looking to make happen isn't viable.
Keep using your existing system for as long as you can bear to, then buy/ build a system that meets your needs (and doesn't have proprietary connectors).
Just a note on buying 2nd hand PC stuff. There are, I'm sure, some genuinely good deals out there. But it's very, very easy to end up sabotaging yourself by buying 2nd hand - often the pricing is basically a trap, you get less or simply 0 warranty, and you limit yourself for future upgrades meaning you need to spend more in the long run to replace things just to let you upgrade the stuff you want to upgrade.
When you do replace your system, give very serious consideration to buying new.