Few points:
Mary's legitimacy - was not in question at the start of her father's attempts to get out of his marriage to Catherine. Had his request for an annulment been granted it would almost certainly have preserved her legitimacy (though it would be open to question in future which naturally concerned her mother).
Catherine's primary reason for arguing her marriage was valid was that she was a devout woman who believed it to be so in the eyes of God. She probably believed it would imperil her soul to accept Henry's arguments and it cost her everything including her daughter's legitimacy. It is for that reason that when offered an easy way out such as embracing a religious life (freeing Henry to remarry) she consistently refused.
Given she fought and ultimately Henry had to move to get his marriage dissolved in England without Rome's approval Mary became a key concern. Catherine and Mary's refusal to accept the end of the marriage, Mary's illegitimacy and Anne and Henry's marriage, simply hardened Henry's attitude.
Mary (legitimate or not) posed a significant risk to Henry and Anne and any children they might have - she was the cousin of the Emperor, nearly of marriage age and had a lot of supporters at court and in the country at large - more than enough to mount a claim to the throne in the event of Henry's early death it is worth remembering he was in his 40s by this period and every chance he could die before any of his children with Anne were of age.
All the treatment of her at this point was about emphasising her status as the King's illegitimate daughter and not a claimant to the throne, and also about the King's attempts to discipline in his view an unruly and disobedient child - certainly he and Anne could have been kinder, but Mary wasn't just a child she was a symbol of continuing opposition to Anne and a recalcitrant one at that.
The nursemaid idea is a myth - rather similar to the suggestion that Catherine was reduced to absolute penury - Henry took action to reduce their households to ones appropriate for their positions as Dowager Princess of Wales and the King's illegitimate daughter. Catherine's income was still very large.
Mary's personal household was not disbanded until late 1533 - when the King decided his daughter's would share a household - Mary was not a servant in Elizabeth's household, she had her own rooms and her own staff, but her presence was designed to make it clear she was not the heir and would always come behind Elizabeth. The contrast between the two to visitors was obvious.
Henry and Anne's reasons were obviously to elevate Elizabeth's status to the detriment of Mary - its worth remembering that Parliament in 1534 did not do entirely as Henry wished in terms of the first Succession Act - it made clear Elizabeth was heir, but it did not specifically exclude Mary as the King had wanted and in 1536 the leaders of the Pilgrimage of Grace wanted Mary's rights guarding and advancing over Elizabeth's - Mary's status was still up for debate even if the King made it clear.
Catherine's death - Henry's reaction is supposedly to have been one of relief and he did wear yellow and appear to be happy (yellow has been a colour of mourning in some cultures but there is no evidence i've seen that it was used as such by the Spanish royal family at the period) - he is believed to have considered her death would mean a restoration of normal Anglo-Imperial relations. On Anne's reaction there is nothing contemporary about it.
Her celebrating the death would have been a mistake and she was intelligent enough to realise that a) she'd failed to deliver a son and though pregnant would miscarry shortly after Catherine's death b) That Henry was tired of her, his eyes were wandering and she was not the meek submissive wife he preferred c) she was the only remaining impediment to Henry normalising his relationships abroad etc