여전히 (still, just like before) indicates a continuity of (/no change to) an action or behaviour
나는 여전히 뛴다 = I still run (= Just like before, I carry on running or continue to run)
 그래도 (= 그리 하여도: That may be so; Even so, still, nevertheless etc.)
그래도 나는 뛴다 = Still, I run (Even so, I run)
그래도 is an adverbial conjunction and can head a sentence.
여전히 is a time adverb used to modify the verb or the predicate part of a sentence; therefore it is ideally placed before the verb or after the subject.
To head a sentence, (syntactically) 여전히 would need to follow some adverbial conjunction s.a. 하지만 or 그래도, i.e.
하지만 여전히 = But ... still; or
그래도 여전히 = Even so ... still
그래도 여전히 나는 뛴다 = Even so, I still run or Even so, I am sill running.
Today I asked my Korean teacher. He said 뛰다 is a broad term meaning to jump or to run, and he acted out jumping and running. He said 뛰다 includes the verb 달리다 as a subcategory. He said we have to go by context. [The problem is that all these sentences have no context. hahaha!]
March 5, 2020.
The interpretation of 뛰다 as to run or to jump usually is defined by their combinations with some other verbs (뛰어-V) or sometimes by the use of postpostions.
뛰어-오르다 = jump up
뛰어-넘다 = jump over
뛰어-다니다 = run about
뛰어-오다 = come running
My wild guess is DLG want to point out that colloquially, when 뛰다 is used on its own and out of context, it is a short form or synonym of 뛰어가다 (to dash).
뛰야해~!! = Must dash !!
Any feedback on this issue is more than welcome.
다 has various meanings.
In the case of korean verbs, it is used to show their dictionary forms (i.e. the way they appear in dictionaries). It is similar to the "to" in English verbs (but not the same).
e.g. 뛰다 - (to) run
In a sentence, -다 gets dropped off to be replaced by the appropriate verb endings i.e. after verb type, tense, mood, levels/styles of speech have all been taken into account. [Verb conjugation].
In such instance, -다 may reappear but only as part of those verb endings.
2 most obvious styles with -다 ending are:
(1) in plain style (declarative mood):
present tense: -다/ㄴ다/는다
뛰다-> 뛰+ㄴ다 -> 뛴다
past tense: -었다/았다/했다
뛰다-> 뛰+었다-> 뛰었다
future tense: -ㄹ 거다/을 거다
뛰다-> 뛰+ㄹ 거다 -> 뛸 거다
(2) in high formality style (business) with the : -ㅂ니다/습니다 ending. [Declarative mood]
present tense 뜁니다
past tense 뛰었습니다
future tense 뛸 겁니다
Hope I am not confusing you further. Always best to keep a grammar book handy.