This question generated much more of a response than I was expecting, so I thought it would be useful, for me if for no one else, to post what I thought were the most important takeaways. This is going to be too long for a comment so I'm putting it in an answer, though I think at this point the original question has already been answered sufficiently.
We have a situation where there seems to be more than one "phrase" or "idea" or "thought" placed in front of the verb, which seems to violate the verb second (V2) rule. There were several explanations offered, but I think the most important thing going forward is that each explanation can be generalized, and each generalization is set of circumstances in which the V2 rule might seem to be violated but is, in fact, followed.
An enumeration is when multiple phrases of the same type are strung together. For example "Thomas, David und ich" is an enumeration of three nouns. In English, enumerations of nouns normally require an "and", and apparently German requires an "und", but this is optional for adjectives and adverbs. The first takeaway is:
An enumeration counts as a single phrase for the V2 rule.
Thomas, David und ich gehen ins Kino.
Langsam, bedächtig, vorsichtig schlich ich in ihr Zimmer.
Fröhlich, mit Anna an meiner Seite, spaziere ich die Straße entlang.
In the last example, the second phrase is also analyzable as a parenthesis, an explanatory phrase inserted into a sentence, and this interpretation is strengthened by the commas on either side. This interpretation would be clearer with different puntuation:
Fröhlich - mit Anna an meiner Seite - spaziere ich die Straße entlang.
In any case, this brings up the second main takeaway:
Parenthetical phases do not affect the word order in the rest of the sentence, so they do not affect the V2 rule.
The English Wikipedia article on Parenthesis has a good selection of examples, not all of which are relevant here, but I'm going to try to invent my own with the parenthesis in bold:
Nina, meine Schwester, ist verheiratet. (This is an example of apposition.)
Ich, nicht Karl, liebe Maria.
I have the impression that the function of parentheses is often handled differently in German than it is in English; German tends to use relative clauses where English uses present participles in parentheses.
This brings up the third takeaway:
Relative clauses and certain other clauses attached to another phrase do not count for the V2 rule.
Hanna, die sehr groß ist, liebt mich leider nicht.
Mein Vater, der sehr wütend war, stürmte aus dem Haus.
Naomi, wenn sie mich sieht, läuft weg.
Again, the last example might be analyzed as a parenthesis. In such cases where multiple interpretations are possible, it doesn't actually matter since the result, multiple phrases combining to form a single phrase, is all that counts when it comes to the V2 rule.
I'm just a learner so my examples may not be entirely correct, if not then please let me know or just fix them. Hopefully the rest of the information is correct and useful.