The IE-to-Uralic loanword transfer phenomenon is quite messy. There are those basic vocabulary terms that look alike, but it goes beyond that sometimes. The conjecture that the borrowings are borrowings is moderately well accepted, though people have been pointing out the exact same problems you just did: they are so fundamentally basic it's almost silly.
Besides "give", "do", "woman", "person", there's also "wet/water" (PIE * wed- and * woder- corresponding to PU * weti), "fish" (PIE * (s)kʷálo- (whence whale, squalus) corresponding to PU * kala), "to stab, push" (PIE * neiǵʰ- corresponding to PU * nikka) et cetera. Sure, these range from really basic to somewhat peripheral, but there are some words that look very related but are extremely unlikely to be borrowed, such as the first (PIE * me, * méne (acc. and gen.) and PU * mun, * minä (just nom.)) and second person pronouns (PIE * tū́ and PU * te, tun, * tinä/cinä(?)), the demonstrative pronouns (PIE * so-, * to- and PU * sä, * tä, * to), a couple of case endings (PIE * -m, * -od (accusative and ablative) and PU * -m, * -ta/tä (same cases)), as well as some conjecture relating to some other endings that might look related (PIE 1SG/PL and 2SG/PL endings * -m, * -me, * -s/tHa (active and perfect forms), * -te, corresponding to PU * -m, * -me, * -t and * -te)
It really looks like you'd have to strain to say all those were borrowed, though I admit that may be my opinion. Here in Europe, there's some small to moderate mainstream support for this Proto-Indo-Uralic thing: mainstream linguists such as Frederik Kortlandt and Robert Beekes say that it is possible that the two language families are related.
Now, I cannot answer your second question as it's not something I'm familiar with, but regarding the third one, I can say that it is extremely uncommon for basic vocabulary to be borrowed, core morphology even more so. There are a plethora of Indo-European borrowings in Uralic that entered the language(s) at various points in time, sure, but borrowings and loanwords can also stand side-by-side with cognates.
(I think I'm going off on a tangent now, so I'll stop here; I hope the answer was good enough).