Mixing Past & Present Tense

Using future conditional to express something that happened in the past is fine, though here it sets up an expectation of destiny -- as in, these two people are super important and I will tell you more about them later. Your editor was probably trying to simplify the sentence for you. I disagree with their addition of "and" because you end up with a run-on sentence.

Here's my suggestion:

These two peers became my support system later, assisting me when I struggled with the structural engineering portion of my internship project at the lab.

I changed "as I struggled" to "when I struggled" because "as" indicates a more ongoing experience while "when" is more occasional.

If they helped you while you were in the lab, then:

These two peers became my support system later, assisting me in the lab when I struggled with the structural engineering portion of my internship project."

But if your entire structural engineering portion of your internship project was based in the lab and these two peers helped you outside the lab, then leave as I wrote in the first suggestion.

Apparently, the reason you were advised not to use "would" concerns style. From CGEL it can be gathered (§ 4.48 p. 218) that the future time in the past as signified by "would" has the stamp of a "rare, literary narrative style" whereas what is expected in your document is essentially a factual communication.
The present participle "assisting" is the verb form of an adverbial participial clause; from "The Meaning of the English Present Participle)" it can be inferred that an idea of simultaneity tends to be communicated by this construction (between the assigning and the becoming a support system).

[…] the idea that participles express simultaneity accurately characterizes most participial uses. However, Jespersen‘s observation that this simultaneity is "vague" (2006 [1933]: 197) calls for further specification, as it (correctly) suggests that not all participles express simultaneity in the same way and even that some might not express simultaneity at all. For example, the participle "bursting" in (2) is not strictly simultaneous to the higher clause process "leave in its wake":

(2) a bursting bubble can leave in its wake a ring of smaller bubbles (Google)

The idea that participles express progressive meaning is more problematic. Quirk et al. (1985), Declerck (1991a) and Huddleston & Pullum (2002) all take pains to qualify this generalization, recognizing that many participles are not progressive in meaning, but they do not explain under which conditions the progressive meanings arise. Only Declerck (1991a: 449) suggests a rule when claiming that it is the participles of dynamic verbs that are progressive, but this quickly turns out to be inadequate as a generalization.

The reason this past participle has been discarded by your corrector is the lack of clear identification of the concept: for one, it is not clear whether the process of becoming a support system was a point-wise happening in time or a gradual event during which the assisting could be conceived as simultaneous, whereas in the former case simultaneity leaves an impression of incongruity. As this participial clause can't be related easily to the matrix clause it makes for a construction that does not communicate logical relations clearly, while the changes you have been advised to make remedy that problem. It follows that the past participle has not been eliminated because it shouldn't communicate the right notion of tense but because of the uncertainty of the syntax.

The addition of "and" is explained simply: you have initially a clause that is connected by a comma, and it is an adverbial clause, a so called supplementive clause.

From CGEL, § 15.60, p. 1123

"when adverbial participle clauses and adverbial verbless clauses are not introduced by a subordinator, there may be considerable indeterminacy as to the semantic relationship to be inferred. […] In their indeterminacy, adverbial participle and verbless clauses resemble the versatile relationship expressed by nonrestrictive clauses and the connective function of the coordinator "and". They are all capable of assuming, according to context, a more precise role:

• Jason, told of his son's accident, immediately phoned the hospital.
• Jason, who was told of his son's accident, immediately phoned the hospital.
• Jason was told of his son's accident, and he immediately phoned the hospital.

For all three sentences, although the mode of clause connection does not make this explicit, we infer a temporal sequence of events.

It is not clear to me why "me" was added since "assist" is intransitive with the same meaning; perhaps, the idea was to add some specificity, which seems to be preferable even if not absolutely necessary.