"For kings" - they were accustomed to date their contracts by the year of the king from the year of the king's ascension as is said in Tractate Gittin (80a) to have a peaceful relationship with the kingdom. The Sages established the first of Nissan as the start of his year. Even if he ascended to the throne in Shevat or in Adar his [first] year ends when Nissan comes and they start to count the 2nd year.

And for the festivals - This is explained in the Gemara (4a)

For animal tithes - Because you cannot give tithes for one year's newborns from another year's as is written (Deuteronomy 14:22-23), "You shall surely tithe all the produce of your field each and every year" and an [anonymous] master said in tractate Bechorot in the chapter Animal Tithes (53b in Bechorot) "[Since the verse uses 2 words to describe tithing when one would've sufficed] the verse is speaking about 2 tithes. The first is animal tithes and the second is grain tithes

For years - this is explained in the Gemara (8a)

For Sabbatical and Jubilee [years] - when Tishrei comes it is forbidden to sow and plant from the Torah

For planting - For the Orlah counts and even if it was planted in Av its first year ends at the end of Elul and the Gemara will explain the reason for all of this.

For vegetables - For vegetable tithes that you don't separate and tithes [after having mixed] those collected before and those collected after Rosh Hashana

For trees - This is talking about the fruit of a tree that was formed prior to that date belonging to the previous tithe year and thus cannot be tithed together with fruit that was formed after that date. In the Gemara it will be explained why Shevat is different.

Gemara, with regard to what halacha - That is to say why is there a separate day to count for kings perhaps every King's year should begin from his ascension date [meaning, if king A took over 1 Shevat and King B who took over after him took over 2 Adar the next year perhaps the new year for Kings should be reset from 1 Shevat to 2 Adar in accordance with when the king took power]

For contracts - To figure out whether a contract was post or antedated

As was taught in a Mishna - in Tractate Sheviit (10:5)

A year is counted for him - That is to say his first year ends and from then they begin counting the second year