[At the moment, I focus on the evidence relating to a eukaryotic origin and will in a later stage put some more effort in disputing the endosymbiotic theory. As I have discussed here, the endosymbiotic teory is not substantial enough for me to be considered a scientific theory, while major hurdles in the presumed mechanistic scenario give enough reason to look at alternative theories, sepcifically a eukaryotic origin.]
1. Mitochondria have the appearance of bacteria?
It has been claimed that mitochondria resemble bacteria (here) and in virtually all textbooks the mitochondrion is represented in the shape of a bacterium (here). In reality, it is hard to find fluorescence images of mitochondria in eukaryotic cells that look like individual spheres. Instead, mitochondria seem to form a dynamic reticulum with interconnecting tubules that resemble the eukaryotic membrane system (here), instead of a bacteria such as Rickettsia (here). Many of the classical electron microscopy images of mitochondria represented various cross sections of the mitochondrial tubules giving the false impression of a sphere (see for instance here). The endosymbiotic theory was formed based on images like this, but do you think that it would have had any chance when we has access to images and 3D models of mitochondria such as here, here, here or here)?
It is not only my inference directly from the pictures , but current literature on mitochondrial morphology acknowledges that mitochondria should be seen as dynamic reticular structures (here).
I would say that the real appearance as a reticulum with interconnecting tubules poses a problem for the endosymbiotic theory, because it somehow has to explain in quite detail how we get alpha-proteobacteria such as Rickettsia to evolve into a reticular membrane structure. Also, we would at least expect some organisms that have conserved the bacterial morphology in their mitochondria, as they should have been fruitful partners for tens of millions of years before evolving into reticular structures.
2. The mitochondrial membrane resembles a bacterial membrane?
Mitochondria have a double membrane and the endosymbiotic theory supposes that the original membrane of mitochondria was bacterial in nature. It is not entirely clear, however, whether only the inner membrane or both outer and inner membrane was of a bacterial nature. If we look at the illustrations here, we see two different scenarios in the drawings, a single-membraned (e.g. here, here, here) versus a double-membraned (here, here) bacteria as a host. Although the difference seems not to be relevant to most textbook editors, they do represent two different scenarios of which one, the classical picture of the endosymbiotic event by engulfment is Lamarckian in nature and contradicts genetic determinism (here).
The bacterial membrane is one of the basic characteristics that distinguish bacteria from eukaryotes, see some examples here. In order for mitochondria to resemble bacterial membranes, they should share characteristics such as a cell wall with peptidoglycan and lipopolysaccharides, gram-staining and antibiotic sensitivity. Some effects of antibiotics have been seen with both bacteria and mitochondria, but the effect is minor while the use of antibiotics is based on the principle that they distinguish between bacteria and eukarytes, including the mitochondrion (here). Until then, the selection of a few apparent similarities while ignoring the many differences does not indicate a bacterial origin for mitochondria. On the contrary, the fact that their membranes are so different as well as the fact that nearly all genes are encoded by the nucleus is primarily evidence against a bacterial origin.
Even though some shared characteristics may be found, we have to realize that bacterial and eukaryotic membranes are fundamentally different. It seems virtually impossible to change all fundamental bacterial membrane characteristics and replace them with a eukaryotic counterpart without loosing membrane integrity. The differences between the membranes of mitochondria and the cell walls of bacteria make the endosymbiotic theory mechanistically difficult. It seems quite clear that bacterial membranes do not change easily into other membranes, and frankly I don’t see any scenarios in which to change all these membrane components without drastically affecting fitness.
3. The organization of mitochondrial DNA is bacterial in nature?
There are various claims that use the structure of mitochondrial DNA as evidence in favor of the endosymbiotic theory. The main is that mitochondrial DNA is circular, just as bacterial DNA. This is actually not entirely true since many mitochondrial DNAs are linear and to make things more complicated, bacterial DNA can be linear as well.