3850 series models are all copper. If you need one with fiber interfaces, you would need to go 3750X, (as you already know).
The best you can do on a 3850 is 4 x 1 Gbps SFPs in the optional C3850-NM-4-1G network module.
Besides the 3750X you mentioned, you could also move up to a 4500X which is all SFP-based. That's a much more capable switch though and will take you up a tier in price.
Yes there is. 3850-12S and 3850-24S. Please see the link
The 3850-12S/24-S was released to the public a few days before Cisco Live SF 2014.
Please note that the modular uplinks are only 1 Gbps for the time being. The 10 Gbps uplink will be made available by the end of 2014.
Also note that if you are going to use or add this model to an exhausting stack you need to factor in the minimum required software to support this model.
Please see https://www.cisco.com/c/dam/en/us/products/collateral/switches/catalyst-3850-series-switches/at-a-glance-c45-731636.pdf