Most of the time while performing four rakat prayers, I forget which rakat I was performing, was it the 3rd one or 4th one. In such a state, what can I do?

You assume it is the fourth rakat. And after finishing the prayer with tasleeem, you pray one rakat prayer, you recite in it the Fatiha only then bow and prostrate twice and finish it with tasleem. You can do that while standing or offer two rakat with the Fatiha alone in each one of them in sitting position.

After having been granted the legitimate wish/desire by Allah Almighty, how long more can we continue offering our thanksgiving prayer?

The said prayer is not obligatory, it can be repeated. An individual is expected to feel the Godly bounties that surround him day and night. Such feeling is not limited to the time after granting the wishes. It was reported that it is recommended for an individual to say everyday seven times "Alhamdulillah ala kuli nematin kanet aw heya kainah" which means "thank god for every bounty that happened or will happen".

What would you say about this if the guy who approached me for marriage wants to see me without hijab?

It is permissible for the man to see the woman that wants to marry her. Based on compulsory precaution, what to be uncovered should be limited to what usually shown at home like the arms, the head and part of the chest and not to exceed that. It is also permissible to wear thin cloths that show the size of the body that tells the woman's shape and not to exceed that.