After za can be only instrumental or accusative.
Accusative follows "za" when it means behind, and the verb indicates movement ( go, hide, fall,throw). In figurative meaning you need to know verb+preposition +case
Edit: I deleted the part about figurative meaning.
I guess You need to know verb+preposition+case
There are many figurative examples of za+Acc.:
– płacić za X – pay for X
– karać za X – punish for X
– dziękować za X – thank for X
– trzymać za X – hold by X
– chwytać za X – grab by X
– uznawać/uważać (się) za X – consider (oneself) X
– za X where X is a period of time – in X
– za X where X indicates payment – for X
– przez is always followed by accusative
– przy is always followed by locative
– o, na, w, po are followed by locative or accusative
– few prepositions (e.g. dzięki, ku, przeciw(ko)), are always followed by dative
– several prepositions (za, (po)między, nad, pod, przed) are followed by either instrumental or accusative – these are usually used to indicate location
– z is followed by either genitive (meaning from) or instrumental (meaning together with)
– some prepositions (like do, od, według, dla, wzdłuż, bez, obok, koło, u, wokół, and all prepositions formed by prefixing instrumental-followed prepositions with z- or s-, like zza, znad, spod, spomiędzy, sprzed) are always followed by genitive
– if a preposition (other than z) is used with two cases and has locational meaning, then accusative indicates direction towards, and instrumental/locative indicates location (similar as in German)
– no preposition can be followed by nominative or vocative
Yes it is "(Ja) Tęsknię za tobą."
Unfortunately I do not know anything about conjunction tables, but http://sjp.pwn.pl/doroszewski/tesknic;5507602.html claims it is conjunction VI a, and https://pl.wiktionary.org/wiki/Aneks:Język_polski_-_koniugacja_VIa has conjuction of "robić" as example. (with the same "i"
It also says
Jeżeli ostatnią głoską tematu jest "l" (L), to w czasie teraźniejszym (dla dokonanych: przyszłym) w 1 osobie liczby pojedynczej i 3 osobie liczby mnogiej występuje twarde "l" (np. "woleć" → "wolę", "wolą", ale "wolisz", "wolimy" itd.). To samo zjawisko występuje także we wszystkich imiesłowach oraz rzeczowniku odczasownikowym ("woląc").
If the last sound of the root is "l" , in present tense (future for perfective verbs) in 1st person singular and 3rd person plural there is hard "l. The same happens in all participles and in a noun.
You can find the conjugation/declension of any verb/noun/pronoun/adjective/adverb at sgjp.pl.
I'm not sure what you mean about the extra "i" - the present conjugation of "tęsknić" seems completely regular to me. According to SGJP it's conjugation 6(i+j) which means the imperative forms have an extra "ij" compared to plain 6a verbs (i.e. "tęsknij", "tęsknijmy" for tęsknić vs. "rób", "róbmy" for robić). There are about 850 other verbs in the same class so it hardly seems irregular.