OnePlus One: Multiple issues involving MTP and Google Play on updating to CM13 via OTA
I'll address your 2nd problem 1st, as that is a simpler thing to describe.

I suffer from the same issue on a stock HTC Nexus 9 (6.0, MRA58K). Whenever I connect it to my PC, it defaults the USB connection mode to "Charge Only", and I have to toggle it back to "MTP" from the persistent notification in the statusbar.

AFAIK this is a general problem with Marshmallow, and one of my friends with a OnePlus One with CM13 also experiences this issue. I'm sorry, but we can only hope some enterprising developer corrects this oversight/bug/feature soon in AOSP.

Edit:

From @Firelord's comment, it appears that the Marshmallow default of "USB Charging" is a deliberate feature in Android 6.0. This is highly inconvenient for some users. See the Android 6.0 official change log here. This link is courtesy of @Firelord's answer here.

Now, the 1st problem.

This error of "Google Play Services has stopped" is common after "dirty-flashed" updates. Since the /data partition remains untouched by an OTA update or a dirty flash, the updated versions of the system apps it stores for Lollipop will interfere with the inbuilt, higher versions for Marshmallow which are in the newly-flashed '/system` partition.

Since you use TWRP recovery, I'll recommend wiping /data and clean flashing a full CM13 zip for the Oneplus One with a compatible (ARMv7 Android 6.0 CM13) GApps package zip. Use TWRP's default wipe of /data without /data/media to keep the data on your (emulated) internal SD.

In the future, consider only OTA updating between minor version upgrades, e.g. from 5.0 to 5.1 or 5.1 to 5.1.1, not 5.1.1 to 6.0