15 amp line connects to -> 20 amp line -> 20 amp GFCI -> 15 amp outlet
The previous installer wired the bathroom GFCI backwards
The symptom of a GFCI that shuts off power to downstream outlets but not itself when tested is a classic indication of a classic mistake.  In particular, GFCIs are not symmetrical devices, unlike regular receptacles.  LINE on a GFCI is the power in side, and LOAD is a dedicated output, connected in parallel with the receptacles on the front.  If you try to put power into LOAD on older (pre-self-test) GFCIs, that power source will power the receptacles directly, bypassing the actual switching mechanism in the GFCI.  Then, when you try to run further loads from what are actually the LINE side terminals, those loads are now on the opposite side of the GFCI's contacts from power, so they appear to be protected when you push TEST.
However, this miswiring condition's bad for GFCI's in the long run (tends to fry internal parts), so your bathroom GFCI should be considered due for replacement now.  When you get the new GFCI in, there will be a yellow &quot;FOR WIZARDS ONLY&quot; sticker covering the LOAD terminals.  Do not remove that sticker yet!  Instead, what you want to do is leave that sticker on while wiring the GFCI LINE terminals to power and testing the results using the GFCI's TEST button and a plug-in tester.  Once you know the GFCI itself is correctly fitted, THEN you can unfasten it from the wall, remove the &quot;FOR WIZARDS ONLY&quot; sticker, and hook your additional receptacle to the LOAD side of the GFCI.
15A/20A doesn't matter here, but your mix of bathroom and outdoor outlets does
The good news for the rest of your situation is that 15A and 20A receptacles, whether regular or GFCI, are only distinguished by their faces.  UL and the NEC require both varieties to be able to handle 20A in a pass through configuration (from one set of screws to the other), so your concerns are irrelevant in that regard.  Furthermore, it's A-OK to have fat wire on a small breaker, since the breaker's still the &quot;weak link&quot;, so to speak.
However, you have two problems with what you are doing if you want this to comply with current Code.  First off, the bathroom receptacles need to be on a 20A circuit to start with.  Furthermore, even if this was a 20A circuit, bathroom receptacles aren't allowed to share a circuit with non-bathroom receptacles in any case.  Due to this, to get this up to Code, you'd need to pull a 12/2 for the bathroom, either from your panel as a new home run, or from an existing 20A circuit that feeds nothing but bathroom receptacles.