Howdy madcyclist...
This is my first post (been lurking about a month now). First of all, a bit of personal history, just so you know where I'm comin' from:
I'm male, 52-years-old, who USED TO ride a road bike 4-5x/week here in sunny San Diego. I rode all around this beautiful city for about 10 years...and then experienced serious "burn out". I haven't been out on the road for over 10 years. However, during that time, I've maintained some cursory contact with riding by using "spinner" bikes. The first bike I owned, an old workhorse called a Bodyguard 990, was actually a fairly sophisticated "erg" bike that finally gave up the ghost last year. About 3 months ago, I purchased a true "spinner" bike (Schwinn Pro) and have been using it regularly (5-6/week) ever since.
My impressions?
Number one: Technically, these are not "fixed gear" bikes, at least not in my definition of that term. Resistance is fully adjustable, as I'm sure you know, so you can create myriad workouts. Since the "spinners" are pretty bare bones (i.e., not an "erg"), I think you either have to take a class, or get yourself a HRM (as I have) to track your workouts.
Second, I'm not sure what you mean when you say "on a SPINNER you push the weight versus pulling the weight". I find myself doing both, just as on a real bike. Oh, I'm assuming here that you're either clipped in (as when you take a class), or you have replaced the clipped pedals with the clipless variety, as I have. If so, the action (push/pull...circles with the ankles) is pretty similar to true riding.
To address your other questions:
The advantages to training on a SPINNER are many. If you're strapped for time, like I ALWAYS am, jumping on my SPINNER for 60 minutes is a great way to keep in shape. Another advantage: no problems with bad weather (kind of ironic that I would be listing this as an advantage, living in San Diego. It was about 75 degrees and sunny yesterday

). Still another: safety. San Diego traffic can be pretty scary at times; it's one of the factors that lead to my "retirement" from road riding.
Disadvantages? Pretty obvious one: BOREDOM. A lot of people find the mind-numbing sameness of stationary riding just too much to take for any extended period. There ARE ways to alleviate some of that boredom, and you've alluded to one: training simulations. However, as far as I know, SPINNER bikes aren't made for such a high tech option. In fact, I've yet to find a HRM w/cadence function that works on a SPINNER bike (I've bought - and returned - a Polar monitor because the cadence function was useless...no "fork" on a SPINNER bike). You could, as I said, take a class for motivation, and some people find the comaraderie, music, and instructor-induced variety enough to keep them interested. For me, since I own the bike, I use it at home. In fact, I really don't like SPINNER classes ( I HATE the music...just an old fuddy-duddy, I guess

) and, again, time is of the essence for me, so having the bike in my bedroom is much more convenient than driving to a club, taking a class, and driving home.
All in all, indoor cycling on a SPINNER can be most beneficial. If you're only doing it for bad-weather training - or just for a change of pace - it's an excellent way to stay in shape over the short haul. It can even function as your only "cycling" experience, like it has for me. But, generally speaking, I think I'm more the exception than the rule on this.
One man's opinion. Hope it helped.
Jim