Re: i7 3970X (the most powerful intel cpu from that gen) isn't supported. Is it limited by chipset itself or due to power consumption and weak power supply?

X series cpu require a completely different socket (and different chipset).  This one needs LGA2011.  It is used in Dell Aurora R4.  790 uses main stream socket lga1155.

For RAM I would give it maximum. 16Gb listed but the real supported amount is 32GB I read somewhere. Please correct me.

yes 32= 4x8 GB is doable.  Just make sure 8 GB ddr3 spec is non ECC desktop only 1333 MHz.

For GPU I really don't know. Do you think that nvidia gtx1070 would be ok, if I replaced the original supply for stronger one?

What do you think about upgrading this computer to its maximum?

for gaming no real need to go beyond 16 GB.  32 GB looks better on paper but has no tangible benefit in gaming.  Save the investment for a better gpu.

Cpu is old Sandy bridge Intel 2nd gen which limits performance boost potential.  There is still a lot of game one can play with i7-2600 and compatible gpu, just not as great as some faster cpu/gpu depending on settings/resolution.

A 4th gen cpu/motherboard (e.g., XPS 8700, optiplex 9020) would better match 1070/1080/1080Ti.

Stock psu needs to be upgraded for powerful gpu.  

Whether you have an MT, DT, or SFF, you can power 75w GPU card.  USFF's have no expansion slots.  Statement for DT and SFF's is outside of past Dell specs.  However, YouTubers have been powering 75w cards in DT's and SFF's in the same gen as the 790 for years.  And due to no negative feedback, my answers in regards have the support of moderation, one of whom is a Dell engineer.

Both cards, GTX 1050 and GTX 1050 ti, are 75w.  The 1050 will go best with your i5-2400.  The 1050 ti will go best with an i7-2600.

For future reference, it will help to know what size/form factor your 790 is.