There's no reason not to ensure you get credit for every flight.

IMO, there is relatively little difference between airlines, although some of the LCC (low cost carriers) can be more on the unpleasant side.

Many people swear by Southwest; I don't like them because I don't like to have to line up and grab the seat I want. They don't have first class seats, but they do have a companion pass which I think you need to fly 110,000 miles in a year to get, but when you get it you can take a companion with you for free. (I could be wrong about that.)

Note that most airlines status these days has both a miles flown and a dollars spent component. If you are spending a lot it would be unfortunate to not get status for all that money. As a rule you must spend a certain amount and fly EITHER a certain number of miles or a certain number of segments. It usually works out to 833 miles per segment, so a status requiring 25,000 miles would require 30 segments. A segment is a takeoff and landing, so if you fly from A to C changing planes in B, that's two segments, and if you then fly home that's two more segments.

There is a lot of randomness in getting first class upgrades. However, all airlines have some kind of "premium coach" (distinct from "Premium Economy" on overseas flights) which gives you a few extra inches of legroom, often free drinks and early boarding. Early boarding can be a major convenience if you are carrying on luggage. If you are checking bags, you might want to consider getting airline credit cards anyway, because they all provide for one free checked bag. So a $95 annual fee would pay for itself on two round trips, maybe less (is the first bag up to $30 now? I haven't paid to check a bag since 2013). On American, Platinum elites (50,000 miles and $6000 annual spend) and above get these seats for free, and Gold elites (25,000/$3000) get them for free 24 hours before departure. For a road warrior, those few extra inches of legroom (and, depending on your preferences, the free alcohol) can make the difference between a trip that is miserable and one that is just unpleasant.

You should find out just how your travel agency or department works. And obviously, you should consider the schedules and routes of the various airlines. TVC seems, unsurprisingly, to serve most of the major airline hubs, as well as a few direct flights to Florida. Are you mostly going to be going out the night before? If so, you might have a preference for an airline with a 5PM departure to their hub over a 7PM departure. (Do you want to be able to stay at the office until 5PM, or do you want to leave early and get to your destination before midnight?)

I think you should try to earn status, but it will depend how much flexibility your travel agency gives you.

Two other things to know about. Many airlines have a "status challenge." On American, you can pay a few hundred dollars (usually no more than $250ish) and if you fly, say, 25% of the annual threshold for gold or platinum in 30 days, you will get that status for the rest of the year and possibly the entire next year. Also is the "Mileage Run." Say that it's December and you project yourself to end the year having spend $8000 but flown only 56 segments. It's not at all uncommon (and not against any airline rule) to fly the cheapest possible 4-segment round trip to put yourself over. I did this once, getting up at 4AM to fly from Raleigh to Philadelphia to New York to immediately turn around and reverse the route, getting home in time for a late lunch and getting my 4 segments in.