Slimming the footprint of the case is a good idea, however, from my recent shopping of PC parts, while the case is relatively inexpensive, the boards can get very expensive and quite quickly.
Mini ITX cases can also have some breathing issues, depending on the case.
, micro-atx/mini-itx tower like cases don't have those issues, since they most often are big enough to support an normal/atx psu and big gpu's.
Only with the smaller cases the psu can get expensive and only if you really need a lot of power (>500 Watt). I bought my mini-itx gaming mobo with wifi for 100 euro (including taxes) and for the psu I choose a ~450 Watt gold & modular corsair (80 euro including taxes). All the other parts are similar to atx form factor in price, except for possible the gpu if you choose low profile (which I personally did for the 1050 Ti though it was only 10 bucks more expensive so not to bad
). Concerning heat, it really depends on the case. If it is more htpc or console sized like (/not tower like), I would suggest to stick to good mid-end gpu's (1050Ti, 1060, rx460/470/480, etc) so they don't generate to much heat, and not e.g. put a 1080Ti in it (though most likely that won't fit than anyway