Eating all my day's calories

I think you should.

Assuming you set your MFP profile activity level based on your daily life activity (excluding exercise) as per MFP's instructions, you should log your exercise and eat those calories, too.

The MFP "Strength training (weight lifting, weight training" entry in the cardiovascular section is a conservative estimate for regular reps/sets dumbbell work. It assumes the time logged includes normal between set rests. If what you do is more fast paced, high rep, lower weight continuous work, you could consider "Circuit training, general", although that calorie estimate could be more questionable depending on pace/intensity.

For your stationary biking, if the bike gives you an average watts for the workout. You can estimate the calories by multiplying average watts times 3.6 times hours (it can be fractional hours). That's a reasonable estimate of net calories, which is ideal. You can log it with a custom exercise you create yourself, or change the calories when using an MFP "Stationary bike" entry.

If your bike doesn't measure/report average watts, you can use one of the MFP "Stationary bike" entries, but they may be generous with calories because the different intensities are subjective. The "moderate" ones are consistent with the calculation above for me, for mostly heart rate zone 3 workouts (in a 5 zone scheme), but if you're relatively new to this you may see a higher heart rate for a similar intensity (pace). (I'm F, 68, but I've consistently been doing moderate to intense cardio regularly for around 22-23 years so am somewhat conditioned to it.)

A fitness tracker estimate may be OK, if synced with MFP. Usually, trackers display the gross calories which will overstate your burn (because it includes basal metabolic rate (BMR), your "just being alive" calories, which MFP already assumes you're burning during that time). Some trackers will separately report "active calories" and "total calories" for an exercise session. The "active calories" would be a more accurate thing to log in MFP if logging by typing it in. (When the tracker is synced to MFP, MFP sorts that out for you.)

Yes, eat your exercise calories to fuel your exercise for more effective results. (I ate back all my carefully-estimated exercise calories through a year of weight loss, obese to healthy weight, and nearly 8 years of maintenance since. It works fine for me.)

Best wishes!

I try to just eat back half, in case the calculations are off.