The guards take your stuff as you enter, but they should leave you the letter and axe. Some have had a problem entering with either item, then returning with the second.
You can also toss things into the room, and pick them up when you enter, or kill the guards using an area effect spell that does not need a target (Poison Cloud) so you do not loose any reputation.
You were suppose to give the axe to the king, and just report the matter solved to the dwarven council member (who should have just moved to the house south of the monument).
The following may work to bring Tingalf back, but I have not tried it.
Reviving dead NPCsEven with the dragon shield unidentified, it would still morph. A morphed shield would probably still work to wear the full dragon set once, though, to complete that quest and get the experience / skill reward.