"1. If Cisco has some default bandwidth reservation for control plane traffic (like BGP). Because it looks it doesn't."
Yes, agreed, it appears Cisco does not provide special treatment for BGP.
"2. What can be done in my case? Should I configure QoS on the Router's interface to give BGP packets priority?"
Yes, using QoS above and beyond whatever the device is doing now would, generally, be an appropriate approach. Unsure you really need, so much, to "prioritize" BGP, possibly the need is more to avoid those packets being dropped. Either or both usually can be accomplished via QoS.
One wrinkle to your problem, you're using MPLS. I've never used native MPLS on an interface, and so, off-the-top-of-my-head, don't know whether standard interface QoS works a bit differently when using MPLS. I "know" MPLS experimental 3 bit field can be used for a QoS setting, but don't recall if an IP packet's ToS, by default, maps into it (perhaps using the IPPrec bits) or whether you need to configure a ToS to MPLS experimental mapping.
Further, in your case, I suspect class-default using FQ (as you mention using a router) might solve the issue, but here too, unsure interface QoS would still see multiple flows or consider all the MPLS tagged frames similar to a GRE tunnel, i.e. only one flow.
If the latter, likely you could still treat your BGP traffic "special" (similar to what @Georg Pauwen posted), but again, we need to confirm who/when you convert IP ToS to MPLS Experimental.
BTW, a very important question, is the 10 Mbps a logical cap or the physical interface bandwidth? If the latter, we'll need to shape the egress traffic to 10 Mbps too.
Oh, and to be clear, we need to apply an egress interface policy on the device providing the 10 Mbps data stream.
What's the egress platform and its IOS version?
