In the following sentence:
アカネは菜食主義者だと言っているが、魚は食べる。
それって、筋が通るかな？
Why it uses the てform of the verb 言う and not its ますform?
If I wanted to say the same sentence but in the past tense should I use と言っていた or と言いました, と言った?
What does それって mean? Is it similar to それ？
「アカネは菜食主義者{さいしょくしゅぎしゃ}だと言{い}っているが、魚{さかな}は食{た}べる。 それって、筋{すじ}が通{とお}るかな？」
1-Why it is used the てform of the verb 言う and not it's ますform?
Because Akane has been saying it multiple times (in the past). 「言っている」, in this sentence, does not mean that she is saying it right now as in the present progressive.
「菜食主義者」 = "vegetarian"
The writer/speaker could have used 「言っています」 instead to be on the polite side, but s/he could not have used 「言います」.
「言っている」, as in this case, is often used to refer to the fact that someone keeps saying the same thing as a rather strong personal opinion, belief, assertion, etc. This nuance could not be expressed by a simple 「言う」 or 「言います」.
2-if I wanted to say the same sentence but in the past tense should I use と言っていた or と言いました、と言った?
In order to maintain the same nuance discussed above, you would need to use 「言っていました」 or 「言っていた」. You cannot use a simple 「言いました」 or 「言った」 to create the same important nuance.
3-what does それって means? Is it similar to それ？
This has been answered by @user3856370 in the comments above. 「って」 is a very common informal topic-introducing particle which means practically the same thing as 「は」.
"Very common" is probably an understatement. You could not speak Japanese informally for one minute without using this 「って」 at least once.
