Zooming a pherical projection in matplotlib

Question

I need to display a catalogue of galaxies projected on the sky. Not all the sky is relevant here, so I need to center an zoom on the relevant part. I am OK with more or less any projection, like Lambert, Mollweide, etc. Here are mock data and code sample, using Mollweide:

# Generating mock data
np.random.seed(1234)
(RA,Dec)=(np.random.rand(100)*60 for _ in range(2))

# Creating projection
projection='mollweide'
fig = plt.figure(figsize=(20, 10));
ax = fig.add_subplot(111, projection=projection);

ax.scatter(np.radians(RA),np.radians(Dec)); 

# Creating axes
xtick_labels = ["$150^{\circ}$", "$120^{\circ}$", "$90^{\circ}$", "$60^{\circ}$", "$30^{\circ}$", "$0^{\circ}$", 
                "$330^{\circ}$", "$300^{\circ}$", "$270^{\circ}$", "$240^{\circ}$", "$210^{\circ}$"]
labels = ax.set_xticklabels(xtick_labels, fontsize=15);     
ytick_labels = ["$-75^{\circ}$", "$-60^{\circ}$", "$-45^{\circ}$", "$-30^{\circ}$", "$-15^{\circ}$", 
                "$0^{\circ}$","$15^{\circ}$", "$30^{\circ}$", "$45^{\circ}$", "$60^{\circ}$", 
                "$75^{\circ}$", "$90^{\circ}$"]

ax.set_yticklabels(ytick_labels,fontsize=15);     
ax.set_xlabel("RA");
ax.xaxis.label.set_fontsize(20);
ax.set_ylabel("Dec");
ax.yaxis.label.set_fontsize(20);
ax.grid(True);

The result is the following:

I have tried various set_whateverlim, set_extent, clip_box and so on, as well as importing cartopy and passing ccrs.LambertConformal(central_longitude=...,central_latitude=...) as arguments. I was unable to get a result.

Furthermore, I would like to shift RA tick labels down, as they are difficult to read with real data. Unfortunately, ax.tick_params(pad=-5) doesn't do anything.