I am currently under treatment and take daily medication under the supervision of specialists. I try my best to exercise and live a healthy life with daily remembrance of Allah SWT and the 14 Masoomeen. When I am feeling in good health, I tend to my daily prayers, read Quran, do Zikr, and even read Namaaz e Shab. Unfortunately when I am have days where I cannot get out of bed, even reciting my daily Fard prayers becomes a challenge and often get missed.

Missing salaat really makes me feel worse, but I physically cannot get out of bed. On days I am not well, can I make up my Fard and Nafil Salaat when better?

Also, I want to recite some names of Allah or Astaghfar when I am sick but am unsure if I can do so without being in a state of Wuduu. Could I please get your guidance on this issue?

If your namaz has missed unintentionally either due illness or some other reason, it’s ok but make sure that in Islam all the rulings and shariah are conditional there is no fixed condition specially in Ebadat so for example if there is unusual conditions comes in our life then the ruling will be also accordingly. So if some is sick and can’t do Wudhu or can’t pray while standing or even sitting and can’t perform mustahhabat etc then he /she can perform his/her ebadat on very least level because that time it was his duty was up to that level. So no need to be worry that you can’t pray your salat nicely. Allah doesn’t want our deeds just He wants our commitment. So sometimes if you can pray your namaz by making signs while you have no strength to sit or stand that would be 100 better that you leave your pray and pray it with full strength.
As for as Dua and Asma al Husna concern you may recite even if you have not done Wudhu.