Richard R. Melick Jr (NA Commentary on Philippians, Colossians and Philemon) points out that ὑπάρχω once meant "to exist originally." However, the term later came to signify "really exist" (102). BDAG contains pretty much the same information. It just adds that ὑπάρχω eventually became a "widely used substitute" for εἶναι in Hellenistic Greek "[with] a predicate noun." Moreover, ὑπάρχω occurs quite frequently "in the [participle with] a predicate noun."
I would say that ἁρπαγμὸν carries an active sense in Philippians: better to understand it as "grasping" or "snatching." But see BDAG under ἁρπαγμός and ὑπάρχω.
Keep in mind that words change in time. So while etymologically, ὑπάρχω may have (at one time) meant "under beginning," it eventually came to denote "existence."
I might just add that there is a debate in Biblical scholarship over whether ἁρπαγμός/ἁρπαγμὸν is passive or active (RES RAPTA vs. RES RAPIENDA) in Philippians. If it is active, it would probably imply grasping after that which one does not already possess. The word may also imply a violent seizure (as in a robbery) though it does not necessarily refer to the act of robbery at all times.
(1) Remember that ὑπάρχων is probably a concessive participle in 2:6. So you might render it as "although he was existing" or something to that effect.
(2) ὑπάρχω, by NT times, does not simply refer to a beginning or coming to be.
(3) Although μορφή is anarthous, I would be inclined to definitize "external appearance" since there is only one form of God.
(4) The phrase "taking by force an equality with God" might also be a little awkward. Maybe you should construe ἁρπαγμὸν as a direct object (i.e., he did not consider "a seizure" or "snatching").
The phrase ἐν μορφῇ θεοῦ presents the first crux in our passage. Μορφή (here dat. sg. fem.) is best trs. “form” (most EVV; BDAG 659c). The NIV’s “in very nature God” (“truly God” [CEV]; “God” [NLT]; “possessed of the very nature of God” [H-M 114]) constitutes an interpretation that is neither well supported by the usage of the term in HGk. nor particularly suitable to the surrounding context. Although the term can be used substantially (Plato Phaed. 103e; Resp. 381c; Aristotle Met. 11.1060b; Phys. 2.1.193b; Plut. Quaest. plat. 1003b; Def. orac. 429a; Philo Spec. 1.327–28), there is no semantic component in μορφή that necessarily involves a corresponding “nature” (NIV) or ontology (pace Fee 204; H-M 114). The great majority of instances where μορφή and its cognates occur in HGk. mean simply “outward appearance” (Dan Fabricatore, Form of God, Form of a Servant: An Examination of the Greek Noun μορφή in Philippians 2:6-7 [University Press of America, 2009]; “form, outward appearance, shape” [BDAG 659c]; that “which may be perceived by the senses” [J. Behm, TDNT 4:745-46]).
The comments from the linked blog have been written by J. Hellerman.