Is autolysis a problem when reusing yeast?

Autolysis is not going to be an issue, but viability will be.

In my experience, viability is very hit and miss. I've stored yeast in canning jars in my fridge for 6 months and they still worked, and I've had some that were very slow to get going and some that didn't work at all that were only stored a month.

Really, I don't harvest yeast much anymore. The time, effort and cost that went in to the rest of the brew isn't worth saving a few dollars. In some cases, the flavor profile will not be the same as the original. It's also near impossible to tell how much yeast you are actually storing. Trub has so much non-yeast material in varying degrees, unless you use a microscope and actually count the cells, you really don't know how much you are working with.

On the other hand, it's kind of fun and interesting and you may want to develop a "house" strain.

Ideally, you should probably limit the storage to a few weeks and make a starter at least a week before you brew, in case it takes off slow. You should be able to build up enough yeast in that time and if it goes quickly, you can decant off the starter and put it in your fridge for a couple of days. If you have a backup yeast and use a starter, you'll be able to control the process and not risk your beer.

You'll want to keep the yeast cool (under some beer) or cold (in the fridge after the beer is removed) to minimize autolysis. The warmer it is the faster they run out of glycogen, and once they run out they'll start dying.

Dead cells aren't necessarily a problem, as long as your viability hasn't dropped too much. If most of the yeast are alive, then they will happily eat what's left of the dead ones after they get some sugar and oxygen. However, too much dead yeast is a problem because it means all the other yeast are very stressed. They might not perform as advertised when used again.

Pro brewers usually use yeast 8 to 10 times before they have problems. The problems are usually related to yeast health, but contamination is also a good reason to get fresh yeast from the lab.

Keep in mind that pros are probably storing the yeast only a few days between uses. Maybe someone else can say how long and how many times they get away with at home (no clear answer here). But few serious home brewers do this.