1. Right to marry or remarry is every individual's right, as he/ she may believe that life could neither be full nor complete without having a companion to whom he/she is married in accordance with Law.
1.1. Such a condition if has been imposed in her compassionate ground employment (job) then it violates the prohibition contained in Art. 21 of the Constitution of India.
1.2. Because of such a violation it is liable to be struck down as something contrary to the Constitution and public policy in a welfare State such as that of India.
Summing up; time again various Authorities / Govt. bodies bite dust when they imposed such arbitrary and un-constitutional 'condition' in compasionate ground employement of a widow and in the event of a widow re-marriage post her compassionate ground appointment the challeng by way of Writ were all thrown out by Hon’ble Courts or the State itself corrected its Rules. Hence to your first question I say "it is not illegal to re-marry post seeking job on compassionate grounds by a widow".
2. After the death of her husband she is liable to get share in property including beingnatural guardian custodial share of minor's and are not termed as "illegal".
3. She being natural mother of minor's is having rights to be their (minor’s) natural guardian unless adoption deed with second husband registered otherwise so second husband is also liable to act as natural guardian if registered marriage done in accordance with Law.
I don’t see any prohibition in all your three questions before me.