Learn Finnish: Tässä suunnitelmassa ei ole järkeä.

In existential sentences expressing 'there is' or 'there is no', the subject is typically in the partitive case if the sentence is negative, or if the noun is an uncountable abstract concept. Since järki (sense) is abstract and the sentence is negative, järkeä is strictly required to be in the partitive.

How could I express a similar idea using this lesson's structure (Genitive + olla + Adjective + 1st Infinitive)?

You would need to change the noun järki (sense) into the adjective järkevä (sensible). You could then say for example Minun ei ole järkevää käyttää tätä suunnitelmaa (It is not sensible for me to use this plan). Here minun acts as the genitive experiencer.

Why do we use the inessive case (tässä suunnitelmassa) rather than saying this plan 'has' no sense using the adessive (tällä suunnitelmalla)?

When expressing that an idea or plan makes sense in Finnish, the sense is considered to reside 'inside' the thing. Therefore, you must use the inessive case (-ssa/-ssä) rather than the adessive case (-lla/-llä) which is normally used for animate possession. Saying a plan 'has' sense using the adessive sounds as if the plan is a sentient creature that lacks intelligence.

Is the dictionary form järki considered an old or new -i word?