Is it possible to express both the MHC class-1 and MHC class-2 both in one cell?
Sure generally any cell that expresses MHCII will also express MHCI. Most cells in the body express MHCI as a self-identifier and to present intracellular proteins to T cells as a sort of a status update i.e. the cell will constantly be presenting peptides (pieces of endogenous proteins) on its surface bound to MHCI, if the cell becomes infected it could also present viral proteins on its surface with MHCI, which can potentially be recognized by a T cell specific for that particular epitope. B cells and antigen presenting cells such as dendritic cells and macrophages express MHCII but also need to be able to be identified as self through their basal level of MHCI presentation of self-peptides. The two MHC molecules are recognized by different cell types, MHCI by CD8 T cells and MHCII by CD4 T cells. So you could consider them to work independently from one another. The source of antigens (peptides here) and process for presenting them via MHCI and MHCII are somewhat different from one another.
You should find a mechanism to make this possible - mechanism that controls the ignition of the expression of MHC-I and MHC-II at the same time.
In other words, parallelism is required by the cell. Some think such parallelism exists in one process in childbirth, for instance. I personally think that such processes are in series, than exact parallelism, because such machinery would require large amount of memory, which our cells do not have, and fine tuning machinery, which is would require too much energy to keep to maintain the control system.
In reality, the processes seem to be happening at the same time, because there is so many of them. However, I think this is a paralysis, since different processes are occurring. But to show that two processes happen in exactly the same time would require the fine tuning machinery which should be found first, and also the energy aspect would require too much energy to control it.
Third argument against such parallelism is in probability theory.
I take an example about the logic behind how to falsify your statement.
Statement: If something is a B-cell, then it can express MHC-I and MHC-II at exactly the same time.
Contraposition of this: This something cannot express MHC-I or MHC-II at exactly the same time, then it is not a B-cell. This is wrong because this something is still B-cell, although it cannot express MHC-I or MHC-II in all situations.
Another perspective, the probability that these two events happen at the same time in time space with one independent system is zero. The probability that these two events happen at the same time in time space with two independent systems is zero still. You need to have three systems to make this possible: fine tuning machine and the machines that expresses this. However, I have not seen such machinery to exist. Generally, one point in probability space is always zero.