Is it Haram if my wrist or part of my arm shows while I am praying?
This is due to my sleeve going up when I raise my hands to say “Allahu Akbar”. Also what if one thinks bad thoughts during prayer, does that ruin the prayer?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

All parts of a woman's body should be covered while performing the prayer, since it is an 'Awrah (parts of the body that must be covered) except her face and hands. If she intentionally uncovers some part of her body, her prayer becomes void. So, you have to cover your 'Awrah well so that your prayer will be valid.

As for just thinking about something in your prayer, this does not make your prayer void, but you will lose reward of performing the prayer completely or maybe some of the reward. In other words, this depends on the duration of your state of distraction.

Such a distraction, in fact, is from the devil to prevent you from getting benefit and reward of prayers. Thus, you have to seek refuge with Allah from the devil and do your best to push him away as the Prophet instructed 'Uthman ibn Abu al-‘Aas while the latter came to the Prophet and said: “O Allah's Messenger! The devil came between me and my prayer and recitation and tried to make that obscure for me. The Prophet said: “It is the devil Ghanzab. So, when you become conscious of him, seek refuge with Allah against him and spit on your left side thrice.” ‘Uthman said: ‘I did so, and Allah takes that (insinuation) away of me.” [Muslim]

Here, we state that such an action (seeking refuge with Allah and spitting thrice on left) should be while performing the prayer. This action can be done as follows: turn your head a little to left and spit slightly so that a very little quantity of saliva is expelled.

Allah knows best.