今度はきっと成功するはずです means It should be successful this time.
今度 means this time, きっと is an adverb indicating certainty, 成功する is a suru verb that means to succeed and はず, a noun, which can appear at the end of a sentence followed by だ or です to mean 'to be supposed to (do)'. It is used when the speaker draws a conclusion from preexisting understandings.
Jisho.org shouldn't be relied on for precise usage notes. For example, they say 謙譲 is a な-adjective... but you should use 謙遜 instead for a な-adjective from the 2.
From a quick corpus search:
"謙遜さ" is more common than "謙譲さ"
"謙遜の" is less common than "謙譲の"
"謙遜が好き" is a thousand times more common than "...
This せん is not a negation. そのすべてに感謝せん here means "Let us thank all of them". It's そのすべてに感謝しよう in modern Japanese.
せ is the 未然 (irrealis) form of す ("to do") in classical Japanese, just as し is the 未然 (irrealis, pre-nai) form of する in modern Japanese. ん is a shortening of む, which is a volitional auxiliary in classical Japanese.
-ていく and -てくる are subsidiary verbs that can attach to almost any verb. They can safely attach to 歩く, too.
Difference between -ていく and -てくる
-ていく/-てくる are very common and important in Japanese. Practically speaking, 歩きます without -ていく/-てくる may sound unnatural in many cases, unless you want to emphasize the method of going/coming.
When you feel like using -てくる,...