This is an exceedingly tricky question. Determining the stability of this system is akin to determining the stability of the Solar System, a difficult - and currently unfinished, as far as I know - task. Planetary systems with more than a couple planets are chaotic, meaning that on timescales of ~107 years, they become chaotic and impossible to predict to any degree of accuracy.
If there were simply two planets, I could look for any orbital resonances that would destabilize the system, something that I've written about before. In the cases where $[n:1](2)$ resonances dominate - that is, when the satellites (here, planets) fall into a certain range of masses - then determining stability is (relatively) easy. Here, it isn't.
We can't determine analytically here whether or not the system is stable. I can give you a rough prediction, though, which is that it is stable. Why?
- The planets have relatively low masses (all but one less than the mass of Earth).
- The planets are pretty well spaced apart.
My one concern is that the third and first planets are nearly in a 3:1 resonance, which has the potential to cause instability. But this might not happen. In fact, there's a chance it could stabilize the system even more.
Is the system realistic? I would argue yes. You have three terrestrial planets within about 1.5 AU, which is nothing special. You then have a small gas giant at about 6 AU, a bit further out than Jupiter. Again, this is fine.
What concerns me is that planet way out at 10 AU. It's the least massive of them all - although not by too much - but may still be a terrestrial planet. You have to explain how it came to be way out there. If it formed there, then you need a good explanation; it seems oddly far out. If it was moved back there, you need to explain what caused it to go there. I see one body capable of severely disrupting an orbit, namely, the gas giant. Interactions with the protoplanetary disk and/or the other terrestrial planets could also have had an effect, but those are less likely.
Overall, I'd say you're fine.