when looking for lilly, why did hisao go to shizune's house? How would shizune |or her family| know where lilly is or care?

Lilly and Shizune are cousins, Shizune's dad is Lilly uncle

even though this is not a plot hole, it still does not make any sense. Why did hisao spend a boatload of money for a taxi to go to the airport, instad of taking the train which is faster and cheaper than taking a taxi.

Probably the train does not connect them directly to the airport, but to a few quarters away, so that he would still have to take a taxi

And how would hisao have that much money to throw anyway? It's not like his family is wealthy.

He kinda is, not as Lilly and Shizune though, I put him as Middle/upper Middle class, as you just said in Hanako, Yamaku is a private institution, and these are extremely expensive, even more with the location, structure and professionals that Yamaku has, he even had a pacemaker transplant (which is extremely expensive, costs varies to U$24,000 to U$58,000), besides other cost like annual tuition, uniform cost, application and registration fee, and dormitory fee.  

how does hanako, an orphan, does not have a job, pay to study at yamaku? We know yamaku costs alot of money as it's not a japanese state school, let's not forget that yamaku can't be "for free" as the school needs money and funding since it's not a japanese state school and deals with disabled students, so how does hanako pay for her education?

probably life insurance

again not a plot hole but still does not make any sense. In rin's route Why would hisao smoke a cigarette if he has a dangerous heart condition which can |and will| kill him at any moment? He expresses alot of time how afraid he is of dying of a heart attack, so it is out of hisao's character to do that, as hisao is not |or does not hint| at being suicidal

He's trying to connect with her, so he follows her without paying much attention to the consequences, honestly, Rin route is just too confused to explain