generally a monk could reject certain food and leave it open (of course "risky") to the giver to seek and ask for reason or to try to give else. Bhikkhus are not allowed to ask for certain food for themselves and only if sick there would be certain openings.
to answer the questions direct:
1) What is the proper thing to do if you, as a monk, were to accept alms but have stomach issues so can't eat all the food?
One does not need, is not bond, to eat all what has been given and one can leave it for others, what ever kind of being.
Food which is left should be at least abound on places where are no plants or in water without animals.
2) What would a monk do if they are allergic to the alms given? (lactose and tolerant, alergy)
Generally, since certain well-being out of food is importand for one in training, the Sublime allowed change of location out of reason food for monks, even in the rainy season. If certain food isn't found in an area, not given, or just food that's not enough for ones farewell, one might move on. How much one is bound to certain food is certain a matter of gained liberation, of course. Nevertheless, food, weather, people, dwelling... this are all matters of Upanissaya, strong causes of past, strong developed tendencies, not easy to overcome by common folk.
3) What is the procedure for alms?
Usually, if not a receiver of food in ones dwelling, if an almsgoer, a monk would leave his remote dwelling toward a settlement and walk from house to house with bended head, looking just some meters in front. He may stop word-less at entrances and wait a little (the bowl is covered with the robe in front of him), and if there are signs that someone desires to give, he would then wait. Not really beggging, but giving wise the possibility to make merits actually, is the livelihood of a disciple of the Buddha. When someone approaches to give, he would de-cover the bolw and open the cover a little, looking at the bowl. After that he would walk on, or give an encouragement (some think it's a blessing)/ approve, by a recitation or short teaching. Once he feels that's enough, latest if the bowl is full, he would leave the settlement into remote areas and either return or take his meal somewhere remote. After this he would wash his bowls, after having abound the rest he did not eat, and go on with his usual tasks.
[Note that this isn't given for stacks, exchange, other world-binding trades, but for escape and release from this wheel]