How red are we talking? Because my older brother always gets what i refer to as LAZER EYES when he wakes up from a nap, which are really quite red, although not blood red. If your eyes are literally BLOOD red, then you should probably get that ocular bleeding checked out. These lazer eyes, however, turn completely normal again after 5-10 minutes with no lasting effects, except that i may have poked fun at him about it.
As for post-nap weakness, it's possibly because you wake up during deep sleep, and your body has yet to recover fully from the sleepy time paralysis that affects every non-sleepwalker ever. Besides that, it's normal to not feel super strong after just waking up.
Regarding dry mouth, it could be that you're simply breathing through your mouth while sleeping, possibly because your nostrils are somehow blocked. I know that dries my mouth up at express speed. It could also be dehydration, although that would typically also come with a pounding headache in my experience.
Also, about sleep apnea, Wikipedia tells me that this would rather manifest itself as poor sleep quality rather than what you're feeling, and would not be restricted to naps. If you're worrying about this, you could check with a doctor, although it's not dangerous.
Honestly, i wouldn't worry about this. I also feel super groggy when waking up from a nap, possibly because one doesn't sleep nearly as long or well when napping as opposed to sleeping at night. I am not a doctor, however, so take this with an appropriate amount of salt.