Can we adorn Thirumankappu on the forehead while observing ten day theetu for the deceased? Can we do Gayathri and Sandhyavandhanam during that time? Why should we not touch ulpathiram (vessels used for Perumal) and clothes during that period?

Srichurnam can be adorned in the forehead preferably store-bought and not the consecrated srichurnam. A consecrated Srichurnam is prohibited during these times.
Sandhyavandhanam and Gayathri should be performed but without anganyaasam and karanyaasam.
Theetu means impurity; as the vessels that are used for Perumal should be kept pure hence using them is prohibited during theetu. Theetu is known as asaucam in Sanskrit. As we are impure during a theetu we should not use such vessels.