1. Foreign non-residents may purchase property in Japan.
2. Staying in Japan for 180 days... you are talking about after your current visa expires, you intend to come on a tourist visa (90 days) and extend for a further 90 days? Or you are talking about coming one time on a 90 day visa, leaving once, then coming in at a later date for another 90-day stay? Either way, your entry is up to the discretion of the immigration officer. Coming in and out of the country to constantly renew your 90-day visitor status is something immigration checks with some vigilance. You might be able to get away with it once, but after that I think you will find it difficult to enter the country. (Apologies if you meant something else).
3. Cash purchase is no problem. Typically this is done with bank transfer (and not a sack full of cash handed over) due to the amounts involved. There is no legal restriction on a physical cash transaction, its just a practical consideration.
4. Title registration is typically done by a legal scrivener. Usually you need to provide certain identification documents, (proof of seal registration, juminhyo, gaikokujin registration or certificate of registered items on gaikokujin registration), as well as the inkan which you will use to stamp the documents. Since you are now living and registered here, this is all fairly routine. If you purchase as an overseas resident (non-Japanese) it adds a level of complication.
5. Property tax bills will come once a year. You can pay all at once, or you can split it into 4 quarterly payments. Usually payable at a bank, or at a convenience store in some locations. It depends upon the tax authority. I have had property tax bills from two different locations, and neither of them had a direct debit option, although there is an online payment option... meaning once the bill comes to me (via snail mail) I can pay online.
I have done what you intend to do. In my case the real estate agents and scriveners, former owners, etc.. were all decent people, and the experience was not particularly traumatic. I consider myself very lucky. Not everyone has a drama-free experience like I did.