Your eyes are the best judge. Save your money and slab them yourself. The best decision is your own! Good luck.
Re: the Barber; I don't think it's a proof, and possibly a counterfeit. If I am reading the photos correctly (and I could be mistaken!) I am seeing raised metal pimples in the right obv field and on the bust. Some letters in GOD appear malformed. If the "pimples" are real it has to be fake. Maybe the scratches were to "mark" it as such?
You will always get really good advice here on CT. Ultimately it is your choice after receiving the advice. Neat that you are collecting after receiving 1/2 of your fathers collection.
Totally agree with M'Lord, the coins are good for your collection/trade but sending to a TPG i think would be spending $$ that could go to a better use. JMO Semper Fi Phil
I agree, not worth the time and effort to send them in. Cleaned. I also thing the draped bust half looks suspect. The denticles just don't seen quite right. Could be wrong. It wouldn't be the first time. Thanks for sharing.
Flying Eagles - I assume the 1868s are a typo and 1858s are what you meant. Small Cents are usually what is referenced by the term Flying Eagle, but somewhere in the back of my mind, the term was used for a pattern or two, possibly gold. The 1856 Flying Eagle has been a point of debate on whether it was considered a pattern or a General Issue, though it appears that general issue has won the day. The Whitman Folder had slots for 1857, 1858 Small Letters ans 1858 Large Letters when I was young. But I do seem to recall a Private mint flying eagle and it might have been 1868.
Once again, the capped (not draped) half is OK. The high rims are a trademark of that particular die marriage. From PCGS--NOT MY COIN.
To me, and this is what I see/look for...the coin appears lackluster...void of any natural luster...a "washed out" or "flat" look from cleaning(s), exposure to chems and the sort. Kind of like women's hair after they've colored and bleached it too much/often and/or used various chem-rinses, treatments, etc...the natural (key word, "natural") sheen/luster is shot, never to return or be restored. I ain't no hairdresser but just happen to think of that comparison/analogy.
The Barber looks like a proof. Another way to tell is that on a proof, the rims are squared off which yours appear to be. On a circulation strike the rims are more rounded.
Many articles questioning authenticity and statements that the coins have been cleaned. What better way to establish both would be to spend the money and have them certified (or not if fakes). You can then be sure of what you have, and don't have to worry about possible storage problems for the raw coins. The best part is that it is up to you, based on all the input from experienced collectors, on what action to take. Good luck on whatever you decide to do.
FWIW, I do not see any red flags as to authenticity here. They've all got some issues (mostly cleaning) that would preclude a straight grade if they were sent in, but I don't doubt they're real. (Edited to add: there is, of course, the chance I could be wrong on that.) However, there may be some value to sending the Barber half in, merely to certify it as a proof. That's up to you. I believe (and someone will correct me if I'm wrong), that proof sets of the era that coin (the Barber) was struck in, were issued with tissue paper covering the coins, and the sulfur in the paper caused toning which sometimes progressed to coal black over time! So many of the proof Barbers on the market have been cleaned. I had a lovely PCGS PR63 CAM 1898 half from the Benson collection. It was white with Cameo contrast, and it had likely been cleaned, but not harshly enough for PCGS to "details" grade it. They gave it a straight grade despite some hairlines. I have my doubts that the Barber half in the OP here would straight grade, but it does look like a proof to me.
If the Barber is a proof, not sure? I would say that the surfaces have been altered. Something just looks off to me.