Could you please explain to me what "dtd" and "n.d." mean in those
abbreviations?

"dtd" is the standard abbreviation for "Not Dated" We use 'n.d." to indicate "No date," or "Not dated."
Also "n.p.," for "No place," stated.

Also, do you (or anyone else) know why they list him as "Company D/I"? I've been trying to figure out
if he transferred from D to I. D/I means he served in both companies at different times.

lt appears, although not stated, that upon his stint on Extra Duty as a Nurse at Liberty, Va., he returned to the regiment transferred to Company D.

Mar/April, 1863, n.d., General Hospital, Liberty, Va., Co. D, Nurse
[bi-monthly musters were usually conducted the last day of the period, however, due to operations, many times they were held somewhat later. On rare occasions at an earlier date.

May/June, 1863, present with Company I, at Fayettevill, Pa.