Can a shard blade or shard plate be affected by Allomancy and/or be burned or have attributes stored in it?
Yes, shardblade and shardplate could be affected by allomancy.
There are some caveats to this though.
For starters, it would be very difficult to actually do. Investiture resists investiture; and the stronger the investiture, the stronger it resists.
Something like shardblade or shardplate are pretty far into the invested side of the spectrum; according to Brandon Sanderson, they are essentially pure investiture, so using allomancy on them would require you to have a lot of power in the trait. See the following WoB for an example on steel pushing/iron pulling:
Questioner: Could you use steel or iron to Push or Pull off Shardblades or Shardplate?
Brandon Sanderson: Anything that's Invested resists, the more Invested it is the more it resists.
Questioner: Okay, so you could technically--
Brandon Sanderson: Yes.
Questioner: --if it was not charged?
Brandon Sanderson: Well-- Certain objects just have more Investiture and are more purely of the Investiture. A Shardblade's going to be really hard, but it's possible, it's just going to be really, really hard. Even more hard than an absolutely full Feruchemical metalmind because the Shardblade is being created directly out of the Investiture, it's basically all Investiture, it's not a metal that is Invested. It's going to be real hard.
Burning them would be similar; and that is covered well by Adamant in this related question. The important connections to take from that are that shardblades and plate are godmetal, and therefore we would expect them to be able to be used, like how we've seen Lerasium, Atium, Trell's metal, and ettmetal all used in the metallic arts. The difference with shardblades and plate is that they are far more invested, enough that they have sapience, so using the metallic arts on them would require you to have enough power to overcome those barriers that the investiture adds.