I live in NYC and have been a AA Platinum for 5 year. This year I reached Delta Gold and will be reaching Platinum by year end. I abandoned AA for Delta because of the favorable treatment given to this mid-tier elite level as far as upgrades go. AA requires you to use stickers and First is often already full. I have been upgraded on most Delta flights this year with no problems.

Now to the questions.
For the NY market, is CO better than DL or the reverse? CO matched my Gold and I am not sure if it's better to switch to CO and the Star Alliance.
Is CO or DL better with upgrades?
Who has better fares in NY? EWR is slightly more difficult to get to than LGA and JFK for DL but it's nice that CO is consolidated in one airport.
Is elite easier to maintain with CO or DL?
Are award tickets easier to book on CO? DL has watered down their miles and the Low, Mid or High ticket options are a joke.
Which Credit Card is better? Presidents Plus or Delta Reserve?

Probably depends on where you're traveling. Personally, I picked CO, but EWR is the most convenient airport for me, so AA, DL, etc., were really not good options.

The credit card choice is secondary. Might as well get one that matches the program you commit to.

As to which is easier to maintain, depends on where you go and on whose metal. Dl now has rollover miles which might make it easier, unless of course you find 30% of your travel on *A vs. ST.

While DL has improved greatly I still find it impossible to book an award. I have been trying to burn 150K miles with them, unable too. Even though CO has been called NonePass I still find it easier to use.

Upgrades depend on the routes you fly really. As Plat on CO I've been very successful. Gold on DL I have been happy with percentages.

I am used to CO and find them my preferred carrier all around. I have also gotten used to EWR for a NY airport. Not happy with customer service at EWR.
I am really happy with my Chase Presidents Plus MC.

I would say this is a tough call. A lot of it depends on where you fly. If you do a lot of Europe, then CO has better Germany and UK, but DL has better Italy and France. And so forth. Overall, though, please note that CO is--by far and away--the airline with the most departures out of the NYC area. Also, while JFK may be OK for you (I prefer JFK to EWR because I live and work in Brooklyn), you are clearly well aware of the profound state of decrepitude of both of DL's terminals at JFK.
If you've been receiving upgrades regularly on DL, then I would say DL has the edge here. CO has changed its upgrade system over the years and now a free upgrade for a low- or mid-level tier elite has becoming increasinlgy difficult. Please note that this is especially the case for CO flights in and out of EWR.
I don't think there is any possible correlation between carriers and fares. On the other hand, though, CO deeply discounts its BusinessFirst fares (its international premium class, the equivalent of DL's BusinessElite) both in the winter and the summer. Given the high quality of the product, buying one of these discounted business-class fares on CO is one of the great airline bargains going (I've flown CO in business EWR-CDG for $1,200, EWR-GVA for $1,400 and EWR-TXL for $1,100) Please correct me if I'm mistaken but I have never seen any regular discounting of this depth of DL's TATL business class product. Elite is easier to maintain with CO, since its highest tier only requires 75k miles. But the real question is what good will it do you? If you're often unable to receive upgrades, and award tickets prove to be as hard to snag on *A as they were when CO was ST, then it's pretty much just a plastic card in your wallet to stroke your ego. The advantage does probably go to CO here, especially with their entry to *A

Continental has yet to lose any of my checked bags. Delta, however, has demonstrated expertise at this task . Also, in my experience CO's on-time performance is clearly superior.

DL is a much easier upgrade out of NYC than CO. As GM, I average >80% upgrades on DL, and that includes a lot of transcons purchased on cheap fares.

As others have pointed out, award availability is pretty lousy on DL, and the JFK terminal is ancient. But, for me, that's just the price of having that first class seat. So it's all your priorities.

I recently made elite on CO, as well, but am not too hopeful about upgrades.

DL is known to be better with upgrades. This will vary with route, but generally speaking, that is.

Fares out of JFK tend to be cheaper than EWR. There are always exceptions, but there's a lot more domestic/lower-fare competition out of JFK.

Gold is 50K on either carrier, so probably just as easy to maintain one or the other.

Award tickets will be substantially easier to redeem with CO, now that CO is in *A and has many more partners with good availability to choose from. It can be done with DL, but it'll take some work.

CO has better F service, but your upgrade chances are typically much lower. EWR Terminal C is much nicer than DL's facilities at LGA and JFK. Award redemption on CO is worlds apart. As you said, award redemption is a total joke on DL. That's by far my largest frustration with them.

In my experience, this has not been the case.

In most cases, I find that fares are pretty much the same, that EWR and JFK largely act as one marketplace.

Obviously, there are exceptions on given routes and given dates, but I find it can go either way.

The only real price difference that I have found consistently over time are CO's discounted BF fares, on which it is possible to fly CO J for approximately one-third the cost of DL's BE.

Generally speaking, it is much easier to get advanced confirmed upgrades on DL. One of the nice things about DL.com is that it displays F UG availability on all flights in your elite status UG window. I usually book all of my DL flights five days out and select the combination of fare and instant UG availability. Just booked a $190 OAK-SLC-DTW for later this week and was instantly upgraded. These days on CO, you usually don't get the UGs until the gate... not very reassuring.

On the other, as noted by others, good luck trying to redeem your DL miles.

I find it's the opposite for domestic flights. It's often difficult to buy a Y seat on Continental's transcons at any almost price if it is within 1-2 weeks of departure. My experience has been that DL beats CO for pricing/availability on almost any itinerary from NYC to LAX/SFO/SEA/PDX/LAS, etc. DL is also better about holding back inventory for last-minute buyers, whereas CO will just "sell out" flights a few weeks in advance.

CO J to Europe is a different story, as you mention.

Regarding CO transcons, no question. But there are a lot of other domestic destinations.

On the whole, I find that prices on CO, AA, DL, and B6, the four biggest carriers out of EWR and JFK will generally be within $15-30 of each other for most domestic destinations.

Sometimes CO is cheaper out of EWR, sometimes the other three are cheaper out of JFK.

Outside of transcon I have not found a reliable pattern for domestic.

I am a CO plat based in NYC and recently accepted a status match on DL. As far as upgrades go, I am 3-3 on DL (plat) and all flights have been between JFK and either LAX or SFO. Two times I was on a low cost "T" fare and I believe the other was "H". You could take 100 CO flights between EWR and LAX/SFO on T-N-L fares and expect maybe 3 upgrades out of 100 attempts as a plat. Take that for whatever its worth.