Your furnace would not run properly if the pipes were not sized properly. Is your furnace giving you problems? The installation manual will have a sizing chart in it. Every manufacturer is different. For the brand that I install, the size seems right.

It appears that the vent is larger than the discharge so that outside air will enter the reservoir as fast as the pump empties the liquid. If you have no operational problems, then it's probably large enough. If the sink has a p-trap and the water in the trap is not sucked out by the pump, the airflow should be adequate.

First off, find out where your pipes are located. If you have no pipes near and above that beam, then it likely was not a pipe. If a pipe cracked from freezing, it will not stop leaking but will continue until replaced. Also, that sort of leak tends to be big, like from a break, rather than a drip. However, A drip may appear at a fitting. Likely, you had a roof leak from an ice dam at the eave edge. (Look up information about ice dams to see how these develop.) This sort of leak occurs about 2 to 3 ft from the edge of the roof (that's 2 to 3 ft up the roof from the rain gutter). The overhang of the roof ( soffett ) is cold. Snow melt from high on the roof runs down and freezes at the gutters and on the overhang. As more melt comes down, the ice dam becomes wider that the underlying water proof sheet that is under the shingles and then wets the roof board. Water drips inside wetting the ceiling a little way from the inside wall. This condition is damaging to the structure. There are three things to combat this condition: Better attic insulation, more roof venting, and a better anti-ice dam sheet under the shingles. The attic insulation and roof venting keeps the top of the roof cold and nearer to outside temperature. This delays melting on the upper part of the roof. The snow has a better chance of being blown off. Also melting should not occur until it is warm enough to melt and clear the gutters. A wide anti-ice dam sheet will give better protection from water running under the shingles. The worst condition is when there is a deep snow on the roof and the weather turns warm enough to rain! That's when a wide anti-ice dam sheet is sorely needed.

What he is talking about with pitch is that they recommend 1/4" per foot slope. In reality you can get away with 1/8" slope just fine but I digress. In that rheem manual jake posted, it appears they recommend a downward slope away from the WH so it would have to slope from the top of the ceiling to the bottom of the rim joist. This is opposite a condensing furnace where you want the exhaust sloped towards the furnace so condensate drains back to the furnace drain. I see in that manual the WH has a drain for condensate so I dont understand why you couldnt pitch it towards the WH. Then he for sure could get the slope to work. Actually if you live in an area with snow accumulation you HAVE to pitch it towards the appliance because outside you have to elbow up and elbow out higher than snow would accumulate. You cant have condensate pitch towards outside in that condition because it will fill the exhaust at the elbow, possibly freeze as well. What type of WH are you looking at so we can look up the installation? No you couldnt share with the fireplace. But now that I look at that rheem manual more I see a lot of flaws..... Page 42 they show what I was talking about above, but dont say to slope back to the appliance. Fail. Then Page 43 they show a vertical installation and say to put a condensate trap on the intake?? It should be on the exhaust... And only the intake gets the two elbows at the termination, not the exhaust. So, I guess the WH doesnt want the condensate going through the unit so they could instead install this condensate trap if you wanted to slope towards the unit. It would fit for sure then.