Is it required to be in a state of grace, able to partake in Eucharist in order to give the OT reading in Mass?
It is preferred to be in the state of grace always whether serving as lector or not. The ministry you provide is not impacted by the state of your soul. Other lectors can chime in, please.
It is highly appropriate that those who perform ministries in the Church are known to be of a high moral character and good standing within the Church. (Therefore, it would be inappropriate for someone whose example might provide scandal to the community -- for example, someone living with their partner outside of marriage, or someone remarried without an annulment -- to perform such ministries.)
However, it is not possible to know, at a glance, whether any given person is in a state of grace, just by looking at them as they stand at the ambo. Therefore, the answer to the question you've asked is 'no' -- the notion of being in a state of grace, per se, is not part of the qualifications. (However, if one is in a state of life that is manifestly sinful, then one should seriously consider whether it is appropriate to enter into such a ministry. It is your pastor's responsibility to address these issues with such ministers.)
I personally would never even enter the Sanctuary if I was in a state of Mortal Sin. The sanctuary represents Heaven it is the holiest place in the Church. When we are in state of Mortal Sin then we are agents of the devil and you could say we represent Hell itself, we should not bring that up into the Holy Place. It would seem very inappropriate to me for any person in a sate of sin to exercise a ministry in the Church even if the graces others receive are not dependant.
I can barely even stand to be in the presence of the Blessed sacrament when I am in a Sate of Mortal Sin so I don't know how someone could get so close while knowing they are in such a wretched state.
It was for a funeral and there was no thought / awareness of the grace issue at the time.
For a funeral, where a family member etc. Is doing a reading, it should be recommended beforehand to seek out a priest and confess if necessary even right on the spot. This should be requested in the kindest way, by the family. I think it should also be a requirement for weddings etc.
Is it required for the wedding couple and participants to be in a state of grace too? This should be, as marriage is a sacrament., perhaps during rehersal, for bridesmaids and groomsmen, readers etc.