I am flying EVA from SEA to DPS. I land in TPE around 4:40 AM and leave at 10 AM. Is that enough time to go out into TPE? It seems early, but never having been there some suggestions are welcome.

Assuming you'll catch the first MRT, you'll get to Taipei Main station at 06:34
You should probably get back to TPE airport by 08:30, which means taking either the 07:45 or 08:00 "express" MRT from Taipei Main Station
80 minutes - barely enough time to walk to Fuhang Soy Milk and their long morning queues - but you'll get to stretch your legs and have a Taiwanese breakfast experience

Or if you were to take a taxi into town around 5am you'd be there by 5:45 at the latest, would cost around 1200NT. You could then take the express MRT back per the above poster's suggestion.
As to what you'd do at that time, there's probably not much going on except for the above-mentioned breakfast. You could certainly just walk around the main station area to get a look at the city. Also be aware that the airport MRT station is around a 5-7 minute walk west of the actual main station/breakfast shop (closer to Beimen station) so factor that in.
If Taipei is too much effort you could also enter Taiwan and go to the basement of Terminal 2 where there's a food court, or the 5th floor where there's an observation deck open from 06:30, convenience store and other restaurants (not sure if they'd all be open at that hour). There's actually one observation deck at each end of terminal 2 departures, one facing each runway.

Thanks. I am not sure if i will run out into TPE or not. It might make sense, but I am not sure if I will be jet lagged. We leave at 11:50 PM and arrive 13 hours later. I might just be wide awake.

If I stay in the airport on transit, I would get lounge access? I am *A gold (1k on united) and I am flying on a D fare (business class).

also, the return has a tight connection. 1.25 hours. I assume that is reasonable? Do I have to clear security or something?

You will have access to BR lounge. I don't think I would head out cause there is nothing going on till maybe 10-11a.

Get to the lounge, take a shower, eat something, nap.

1.25h is doable. Yes, you'll have to clear security on the return (only).

a lot of times, the ground crew will come and rush you to your flight if your flight comes in late. if your taking BR 395, they even give u a special tag so ground crew will notice and get you to your flight.

Honestly I will not head out because Taipei is a "late rise" city. So most restaurants and shops will not be opened at those early hours.
If it is 4pm to 10pm, it will be different because you can take MRT and go to the outlet mall in Taoyuan, which are just a few stops away from the airport stop.
Unfortunately those early hours mean that spending some times in the terminal will be best.

EVA Air lounge is not the fanciest, but at least you can take a shower and have a space to rest. The lounge will be busy for the first half of your stay, but will quiet down after the morning bank flights have departed.

While Taoyuan airport is no Singapore Changi, it is still okay. Do some walking along the B/C corridor will be good too, after a long flight.

Carfield

Fuhang Soy Milk

The queue for that place can be crazy crazy long so you very well could get there and not get through the queue.

Thank you to all for the suggestions. I think I will just hang out in the airport.

One other question - I am connecting from a different airline on a different PNR in Seattle. I have a choice - 5 hours in Seattle or 1.5 hours in seattle. I would prefer to do the 1.5 hour connection. Will I get a boarding pass so that I can just walk through the airport in seattle and get on the EVA flight?

On separate tickets that is hugely risky, you need to consider your plan B if your first flight is delayed

I'd certainly roll the dice on that shorter transit myself, BUT I'd be fully prepared (i.e confirmed space on a later flight) with the Plan B as well.