This first edition of OED includes the noun sense of 'gyp' as
2. U.S. slang. A thief.
1889 in Century Dict.
but makes no mention of any relation to gipsy/gypsy (or anything else).
Later editions include a 3rd sense of 'gyp'
3. A fraudulent action; a swindle. Also as adj. Cf. gyp v. orig. U.S.
gyp, v. orig. U.S.
To cheat, trick, swindle.
1889 in Cent. Dict.
And so to the Century Dictionary
Gipsy, Gypsy (abbreviated a lot) [...] They pursue various nomadic occupations, being tinkers, basket-makers, fortune-tellers, dealers in horses, etc., are often expert musicians, and are credited with with thievish propensities. [...]
Which I think rather sets the tone, for the introduction of gyp.
gyp (jip). n. [In the first sense said to be a sportive application of the Greek [greek letters omitted] for vulture, with ref. to their supposed dishonest rapacity; but prob. in this, as in the second sense, an abbr. of gypsy, gypsy as applied to a sly, unscrupulous fellow]
1. A male servant who attends to college rooms. (the OED agrees with this citing "1805 H. K. WHITE in Rem. (1819) I. 209 My bed-maker, whom we call a gyp, from a Greek word signifying a vulture, runs away with everything he can lay his hands on." however Grose 1881 Dictionary has the same sense but derived from wolf)
2. A swindler, especially a swindling horse-dealer; a cheat. Philadelphia Times, May 27 1880. [Slang.]
It might be worth noting that The Century Dictionary mentions gipsery/gipsyry as
A colony of Gipsies; a place of encampment for Gipsies.
Near the city [Philadelphia] are three different gipseries where in summer-time the wagon and the tent may be found. C.G. Leland, The Gypsies.
I don't think Philadelphia is currently a major concentration of Romani/Gypsy but the timing of the documentation of gyp to mean thief, swindler coincides quite neatly with the abolition of Romani slavery in Romania (1864) and the subsequent immigration of Romani peoples to West Europe and the east coast of the USA.
So while The Century Dictionary doesn't explicitly say gyp is definitely derived from gipsy, it does all look like it's pointing that way unless one were to imagine that Philadelphians of the time were acutely aware of the goings on in British universities.
I would think, in general, that most people will assume gyp is derived from gipsy and is therefore a pejorative, racial slur. Whether it is true or not doesn't really matter; once the majority hold that view then it does become a slur, whether that was the original meaning or not. So it can then be used in the way you describe; to say to someone "You're a gyp" means "You're like a gipsy" (using gipsy to mean a type of person you dislike).
The modern chav could also be racial slur should enough people take up the belief that it is from Romany chavi, chavo.
Gype (in Irish and Scottish) meaning a fool. Nothing to do with gipsy at all.
Joseph Wright's The English Dialect Dictionary gives
GYPE, V. and sb. Sc. Irel. Also written gipe Dmf.;guipe Ant.
1. v. To stare foolishly ; to act as a fool.
2. sb. A foolish stare. Enff.
3. A fool, lout ; an awkward, stupid fellow.
Gype seems pretty close to two other Irish/Scottish words which mean the same thing: Gaup and Gawp.
5. sb. A vacant, staring person, a fool, simpleton ; also in pi. form.
6. A stupid, vacant stare ; a wild, anxious look
And those two give Gaupsheet and Gawpshite respectively (Gobshite).
In The Century Dictionary Gipe, Gype means a petticoat, a skirt. An upper frock or cassock
In OED it is given as a tunic (Obs.).
And finally, Pikey.
The Century Dictionary gives (because it doesn't have pikey at all)
piker, n. [pike3 + -er] A tramp; a vagrant. [Slang]
The people called in Acts of Parliament sturdy beggars and vagrants, in the old cant language Abraham men, and in the modern Pikers. Borrow, Wordbook of the English Gypsy Language.
The pike3 is turnpike
OED on Abraham-man (wandering but not specifically Romani/Gipsy)
One of ‘a set of vagabonds, who wandered about the country, soon after the dissolution of the religious houses; the provision of the poor in those places being cut off, and no other substituted.’
OED does have pikey and various definitions of piker.
pikey dial. or slang = PIKER3
1847 J. O. HALLIWELL Dict. Archaic & Provinc. Words 623/2 Piky, a gipsey. Kent.
1874 HOTTEN Slang Dict. 253 Pikey, a tramp or gipsy.
1887 PARISH & SHAW Dict. Kentish Dial. 116 Piky,‥a turnpike traveller; a vagabond; and so generally a low fellow.
1955 P. WILDEBLOOD Against Law 125 My family's all Pikeys, but we ain't on the road no more!
piker3 repeats the 1874 citation from Borrow that is in The Century Dictionary but predates it with
1838 HOLLOWAY Dict. Provinc. 23/2 Cadgers and pikers are tramps. E. Suss.
piker1 is the earliest sense (from 1301), prior to the word gipsy/gypsy/romani/romany, but comes from pick not pike/turnpike.
†1. A robber, a thief; in later use, a petty thief, pilferer;
but by the mid 1500's this had been replaced by picker which we still use in pick-pocket.
Wikipedia does quite a good write-up on pikey at http://en.wikipedia.org/wiki/Pikey but the section about the Medieval Period is almost certainly confusing pykeris with pickers and the 16th Century section is based entirely on Partridge Dictionary of Slang which merely states that pikey derives from C.16 pike: to depart without providing any evidence (although it seems sensible enough, pike was used in that sense from 1400-now, that pikey isn't attested to until 1847 seems a bit of a leap of faith).