As I'm sure you know, four more exoplanets were recently discovered around TRAPPIST-1 (Gillon et al. (2017)), bringing the total to seven — all, amazingly, presumably rocky and near the star's habitable zone. There has been recent work, of course, about the system's habitable zone and whether any of the exoplanets around it could host life, and so I figure I might as well write up an answer taking into account some of the new information we have.
Here's a table of the relevant information about the seven exoplanets (from Grimm et al. (2018)):
$$\begin{array}{|c|c|c|}
\hline \text{Exoplanet} & \text{Mass }(M_{\oplus}) & \text{Semi-major axis }(10^{-2}\text{AU})\\
\hline \text{b} & 1.017^{+0.154}_{-0.143} & 1.15\\
\hline \text{c} & 1.156^{+0.142}_{-0.131} & 1.58\\
\hline \text{d} & 0.297^{+0.039}_{-0.035} & 2.23\\
\hline \text{e} & 0.772^{+0.079}_{-0.075} & 2.93\\
\hline \text{f} & 0.934^{+0.080}_{-0.078} & 3.85\\
\hline \text{g} & 1.148^{+0.098}_{-0.095} & 4.69\\
\hline \text{h} & 0.331^{+0.056}_{-0.049} & 6.19\\
\hline
\end{array}$$
This data is much better than the original measurements by Gillon et al.
For habitable zone models, I'm going to look to Bolmont et al. (2017). First, look at Figure 1b, which models the habitable zone assuming a mass of TRAPPIST-1 of about $0.08M_{\odot}$. I've drawn a green line at ∼500 million years, which is roughly the age of the system:

This means that the habitable zone currently runs from roughly $2\times10^{-2}$ AU to $4\times10^{-2}$ AU. This would put exoplanets d to f within the habitable zone at the moment, with c and g being decent candidates, too. Within another ∼500 million years, the habitable zone will have changed and roughly flattened out, ranging from $1\times10^{-2}$ AU to $3\times10^{-2}$ AU, encompassing b through e. We see, then, that the habitable zone changes over time, as is the case with all stars, and so the answer to your question depends partly on how long each planet will spend in the habitable zone.
The same authors present graphs of hydrogen (the important component for recombination of water) loss and time spent in the habitable zone for stellar masses of $0.01$-$0.01M_{\odot}$, exoplanet masses of $0.1$, $1.0$, and $5.0M_{\oplus}$, and various luminosities:

I'll look at the second panel on the right column, assuming a rocky exoplanet with a mass of $\sim1.0M_{\oplus}$. I've labeled the semi-major axes of exoplanets b through g, and drawn a box from $M_*=0.071M_{\odot}$ to $M_*=0.80M_{\odot}$, the lower half of the mass range for TRAPPIST-1:

From this, it seems that TRAPPIST-1d and TRAPPIST-1e should be in the habitable zone for a while, at least $10^9$ years and probably several times that. TRAPPIST-1f may get $\sim5\times10^8$ years to $10^9$ years, although TRAPPIST-1g will likely spend minimal time there, if any significant time at all.
TRAPPIST-1b and TRAPPIST-1c should be in the habitable zone for a pretty long time, as we saw in Figure 1. However, they will likely lose a lot of hydrogen: perhaps two to four times the mass of the hydrogen in Earth's oceans, in a worst-case scenario (Lower-mass planets may lose less hydrogen, according to the authors' graphs, which is odd. I'll have to look into that.) Having a large amount of water to start with could make this less of a problem, as the authors argue — and they do say that their models probably overestimate hydrogen loss — but it's still a large problem.
Let's also assume that the exoplanets each have atmospheres with some non-insignificant amount of ozone (O'Malley-James & Kaltenegger (2017) believe that this could lead to UV radiation effects no worse than those on Earth). If we also assume Earth-like atmospheres — not much of a stretch, given the mass ranges of these planets — then we actually have decent targets for life.
My top choice is actually TRAPPIST-1e. While its mass is likely much smaller than Earth's (though it's better than exoplanet d), it will stay in the habitable zone for some time (better than f or g) and shouldn't lose too much hydrogen (better than b and c). We can assume, optimistically, it should spend several billion years in the habitable zone.
What was life on Earth doing when the planet was about ∼500 million years old? Well, it was actually just getting started. The first, smallest not-quite-cells-but-still-reproducing life forms had just gotten started, although they hadn't really achieved much. So if TRAPPIST-1e takes a similar path to Earth, life could be here — or almost here.
Fast forward to one billion years. We've got single-celled prokaryotes (i.e. no proper nuclei) which are developing photosynthesis. Eukaryotes are still a ways off, but progress is being made. A bit over a billion years after that, the Great Oxygenation Event happens on Earth, and oxygen levels in Earth's atmosphere skyrocket. Life is probably already on land, and eventually, atmospheric oxygen will help make respiration possible. Animals — even multicellular life, really — is still a ways off.
You're not going to get animal life for several billion years on TRAPPIST-1e, assuming an Earth-like evolutionary trajectory, but I think that if life gets started there, you'll see some animals. It might take longer on TRAPPIST-1b and TRAPPIT-1c, if there's little water, which is widely regarded as leading to the start of life in the oceans (although obviously, life could take other paths; I've been assuming Earth-like life). TRAPPIST-1d might go similarly to TRAPPIST-1e, although lower gravity may change what types of animal life arise, if they ever do (see How does gravity affect evolution of life?); something similar may happen with TRAPPIST-1f. TRAPPIST-1g and TRAPPIST-1h will likely see no significant evolution of animal life; they just won't spend enough time in the habitable zone.
So, to answer your questions
how advanced are native life forms likely to get? Is the development of animal life plausible?
They may become pretty advanced, assuming you pick the right planet (d, e or f, with my top choice being e). Intelligent life is possible, though not guaranteed.
It's been brought to my attention that recent simulations (Wolf (2017)) also suggest that TRAPPIST-1e is the best choice for life. I haven't read the paper yet, but I'm glad to hear that maybe I was right.