Hi! A microscope is sufficient for both uses. To spot dead cells in a yeast culture I usually stain a little amount of cells with methylene blue, which selectively colors only the metabolically dead ones. On the contrary, you can spot lysed yeast cells without staining, they appear like empty, shrunk cell walls, called "ghosts".
if you want to monitor lysis, you centrifuge your samples and analyse protein or DNA content in the supernatant after centrifugation. The values (A 280 for protein or A260 for nucleic acids should come to a maximum when lysis is complete. This is more efficient than microscopy because mechanical lysis (e.g. by glass beads) will completely disrupt some cells. I.e. they will not show up as ghosts anymore.
Hi! Oxonol dye can be used to stain cells and then count them by flow-cytometry. It is a little more expensive than the methylene blue but it reduce errors.
Article A flow-cytometric method for determination of yeast viabilit...
Respiration is not the best method to assay for lysed (and dead) cells, because if mitochondria are sufficiently intact, they will continue respiring even if the cells are disrupted. This is why you can make experiments in isolated mitochondria.
Checking under the microscope after ghosts, or dying with "vital dyes" is a good and fast way for a first intention, but if you really want to know the percentaje of viable yeats, you will need to make a more reliable viability assay as CFU; this is to spread a known amount of cells on YPD (or the adequate medium to your strain) and count colonies after incubation.
Regarding vital dyes, some months ago, I made some experiments with another interesting dye, having good results: FUN1, and besides, you can obtain fun pictures with it!